- Calculate the length of a curve described by y=f(x) from one point to another
- Find the length of a curve defined by x=g(y) from one point to another
- Calculate the total surface area of a solid formed by rotating a curve around an axis
Arc Lengths of Curves
The Main Idea
- Arc length represents the distance along a curve
- Requires smooth functions (continuous derivatives)
- Approximated using line segments, then taking the limit
- Two main formulas:
- For [latex]y = f(x)[/latex]:
- [latex]\text{Arc Length} = \int_a^b \sqrt{1 + [f'(x)]^2} dx[/latex]
- For [latex]x = g(y)[/latex]:
- [latex]\text{Arc Length} = \int_c^d \sqrt{1 + [g'(y)]^2} dy[/latex]
- For [latex]y = f(x)[/latex]:
- Smoothness requirement:
- Function must be differentiable with a continuous derivative
- Derivation approach:
- Partition the interval
- Approximate curve with line segments
- Use Pythagorean theorem for segment length
- Sum segment lengths and take the limit
- Choice of formula:
- Use [latex]y = f(x)[/latex] formula when curve is better expressed as a function of [latex]x[/latex]
- Use [latex]x = g(y)[/latex]formula when curve is better expressed as a function of [latex]y[/latex]
Let [latex]f(x)=\left(\dfrac{4}{3}\right){x}^{3\text{/}2}.[/latex] Calculate the arc length of the graph of [latex]f(x)[/latex] over the interval [latex]\left[0,1\right].[/latex] Round the answer to three decimal places.
Let [latex]f(x)= \sin x.[/latex] Calculate the arc length of the graph of [latex]f(x)[/latex] over the interval [latex]\left[0,\pi \right].[/latex] Use a computer or calculator to approximate the value of the integral.
Area of a Surface of Revolution
The Main Idea
- Surface area is the total area of the outer layer of an object
- For surfaces of revolution, we use calculus to find the area
- The method extends concepts from arc length calculations
- Two main formulas:
- For rotation around [latex]x[/latex]-axis:
- [latex]\text{Surface Area} = \int_a^b 2\pi f(x)\sqrt{1 + [f'(x)]^2} dx[/latex]
- For rotation around [latex]y[/latex]-axis:
- [latex]\text{Surface Area} = \int_c^d 2\pi g(y)\sqrt{1 + [g'(y)]^2} dy[/latex]
- For rotation around [latex]x[/latex]-axis:
- Frustum of a cone:
- Used to approximate small sections of the surface
- Lateral surface area: [latex]S = \pi(r_1 + r_2)l[/latex], where l is slant height
- Derivation approach:
- Partition the interval
- Approximate surface with frustums
- Sum frustum areas and take the limit
- Smooth function requirement:
- Function must be differentiable with a continuous derivative
- Choice of formula:
- Use [latex]x[/latex]-axis formula when curve is better expressed as [latex]y = f(x)[/latex]
- Use [latex]y[/latex]-axis formula when curve is better expressed as [latex]x = g(y)[/latex]
Let [latex]f(x)=\sqrt{1-x}[/latex] over the interval [latex]\left[0,\frac{1}{2}\right].[/latex] Find the surface area of the surface generated by revolving the graph of [latex]f(x)[/latex] around the [latex]x\text{-axis}.[/latex] Round the answer to three decimal places.
Let [latex]g(y)=\sqrt{9-{y}^{2}}[/latex] over the interval [latex]y\in \left[0,2\right].[/latex] Find the surface area of the surface generated by revolving the graph of [latex]g(y)[/latex] around the [latex]y\text{-axis}.[/latex]