Chapter 9: Q. 42 (page 756)

Polar coordinates to sketch the region and evaluate the expressions.
$3 \int_{0}^{\frac{π}{2}} \sin^{2} (3 θ) d θ$

Short Answer

Expert verified

The graph of the region is

The value of the integral is $\begin{array}{rcl} \frac{3 π}{4} \end{array}$ .

Step by step solution

Step 1. Given Integral

The given integral is $3 \int_{0}^{π / 2} \sin^{2} (3 θ) d θ$ .

Step 2. Plot the Region

The general equation to find the area of a region is $A = \frac{1}{2} \int_{α}^{β} (f {(θ))}^{2} d θ$ , where $f (θ)$ is the region.
On comparing the general equation to the given integral, the region is $\sin (3 θ)$ .
Use the graphing utility to plot the function.

Step 3. Integrate

Integrate the given integral as shown below:

$\begin{array}{rcl} 3 \int_{0}^{π / 2} \sin^{2} (3 θ) d θ & = & \frac{3}{2} \int_{0}^{π / 2} 1 - \cos (6 θ) d θ \\ = & \frac{3}{2} (\int_{0}^{\frac{π}{2}} 1 d θ - \int_{0}^{\frac{π}{2}} \cos (6 θ) d θ) \\ = & \frac{3}{2} (\frac{π}{2} - 0) \\ = & \frac{3 π}{4} \end{array}$

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Polar coordinates to sketch the region and evaluate the expressions.
$3 \int_{0}^{\frac{π}{2}} \sin^{2} (3 θ) d θ$

Short Answer

Step by step solution

Step 1. Given Integral

Step 2. Plot the Region

Step 3. Integrate

One App. One Place for Learning.

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Short Answer

Step by step solution

Step 1. Given Integral

Step 2. Plot the Region

Step 3. Integrate

One App. One Place for Learning.

Most popular questions from this chapter

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Polar coordinates to sketch the region and evaluate the expressions.
$3 \int_{0}^{\frac{π}{2}} \sin^{2} (3 θ) d θ$