{"id":180,"date":"2026-01-12T16:00:44","date_gmt":"2026-01-12T16:00:44","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&#038;p=180"},"modified":"2026-01-12T16:00:44","modified_gmt":"2026-01-12T16:00:44","slug":"numerical-and-improper-integration-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/numerical-and-improper-integration-get-stronger-answer-key\/","title":{"raw":"Numerical and Improper Integration: Get Stronger Answer Key","rendered":"Numerical and Improper Integration: Get Stronger Answer Key"},"content":{"raw":"<h2><span data-sheets-root=\"1\">Numerical Integration Methods<\/span><\/h2>\r\n<ol>\r\n \t<li class=\"whitespace-normal break-words\">[latex]0.696[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]9.298[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]0.5000[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]{T}_{4}=18.75[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]0.500[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]1.2819[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]0.6577[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]0.0213[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]1.5629[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]1.9133[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\text{T(4)}=0.1088[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]1.0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Approximate error is [latex]0.000325[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]0.1544[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]6.2807[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]4.606[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]3.41[\/latex] ft<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]{T}_{16}=100.125[\/latex]; absolute error = [latex]0.125[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">about [latex]89,250 [\/latex] m<sup>2<\/sup><\/li>\r\n \t<li class=\"whitespace-normal break-words\">parabola<\/li>\r\n<\/ol>\r\n<h2><span data-sheets-root=\"1\">Error Analysis in Numerical Integration<\/span><\/h2>\r\n<ol>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{1}{7938}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{81}{25,000}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]475[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]174[\/latex]<\/li>\r\n<\/ol>\r\n<h2><span data-sheets-root=\"1\">Improper Integrals<\/span><\/h2>\r\n<ol>\r\n \t<li class=\"whitespace-normal break-words\">divergent<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\pi }{2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{2}{e}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Converges<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Converges to [latex]\\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\u22124[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\pi [\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">diverges<\/li>\r\n \t<li class=\"whitespace-normal break-words\">diverges<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]1.5[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">diverges<\/li>\r\n \t<li class=\"whitespace-normal break-words\">diverges<\/li>\r\n \t<li class=\"whitespace-normal break-words\">diverges<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Both integrals diverge.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">diverges<\/li>\r\n \t<li class=\"whitespace-normal break-words\">diverges<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\pi [\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]0.0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]0.0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]6.0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\pi }{2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]8\\text{ln}\\left(16\\right)-4[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]1.047[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]-1+\\dfrac{2}{\\sqrt{3}}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]7.0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{5\\pi }{2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]3\\pi [\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{1}{s},s&gt;0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{s}{{s}^{2}+4},s&gt;0[\/latex]<\/li>\r\n<\/ol>","rendered":"<h2><span data-sheets-root=\"1\">Numerical Integration Methods<\/span><\/h2>\n<ol>\n<li class=\"whitespace-normal break-words\">[latex]0.696[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]9.298[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]0.5000[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]{T}_{4}=18.75[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]0.500[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]1.2819[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]0.6577[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]0.0213[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]1.5629[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]1.9133[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\text{T(4)}=0.1088[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]1.0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Approximate error is [latex]0.000325[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">[latex]0.1544[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]6.2807[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]4.606[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]3.41[\/latex] ft<\/li>\n<li class=\"whitespace-normal break-words\">[latex]{T}_{16}=100.125[\/latex]; absolute error = [latex]0.125[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">about [latex]89,250[\/latex] m<sup>2<\/sup><\/li>\n<li class=\"whitespace-normal break-words\">parabola<\/li>\n<\/ol>\n<h2><span data-sheets-root=\"1\">Error Analysis in Numerical Integration<\/span><\/h2>\n<ol>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{1}{7938}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{81}{25,000}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]475[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]174[\/latex]<\/li>\n<\/ol>\n<h2><span data-sheets-root=\"1\">Improper Integrals<\/span><\/h2>\n<ol>\n<li class=\"whitespace-normal break-words\">divergent<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\pi }{2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{2}{e}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Converges<\/li>\n<li class=\"whitespace-normal break-words\">Converges to [latex]\\dfrac{1}{2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\u22124[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\pi[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">diverges<\/li>\n<li class=\"whitespace-normal break-words\">diverges<\/li>\n<li class=\"whitespace-normal break-words\">[latex]1.5[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">diverges<\/li>\n<li class=\"whitespace-normal break-words\">diverges<\/li>\n<li class=\"whitespace-normal break-words\">diverges<\/li>\n<li class=\"whitespace-normal break-words\">Both integrals diverge.<\/li>\n<li class=\"whitespace-normal break-words\">diverges<\/li>\n<li class=\"whitespace-normal break-words\">diverges<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\pi[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]0.0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]0.0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]6.0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{\\pi }{2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]8\\text{ln}\\left(16\\right)-4[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]1.047[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]-1+\\dfrac{2}{\\sqrt{3}}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]7.0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{5\\pi }{2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]3\\pi[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{1}{s},s>0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{s}{{s}^{2}+4},s>0[\/latex]<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":7,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":109,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/180"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/180\/revisions"}],"predecessor-version":[{"id":192,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/180\/revisions\/192"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/109"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/180\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=180"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=180"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=180"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=180"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}