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Chapter 16: Problem 7

Evaluate the following iterated integrals. $$\int_{0}^{2} \int_{0}^{1} 4 x y d x d y$$

Short Answer

Expert verified
Question: Evaluate the iterated integral: $$\int_{0}^{2} \int_{0}^{1} 4xy dx dy$$ Answer: 4

Step by step solution

01

Evaluate the inner integral

First, let's focus on the inner integral, which is with respect to x: $$\int_{0}^{1} 4 x y d x$$ To compute this integral, we treat y as a constant and integrate x with respect to the given bounds.
02

Apply the power rule to the inner integral

The power rule states that \(\int x^n dx = \frac{1}{n+1}x^{n+1}\). Applying it to our integral, we get: $$\int_{0}^{1} 4 x y d x = 4y \int_{0}^{1} x d x = 4y \left[\frac{1}{2}x^2 \right]_0^1$$
03

Plug in the limits of integration for the inner integral

Next, we'll use the limits of integration to determine the value of the inner integral: $$4y \left[\frac{1}{2}x^2 \right]_0^1 = 4y \left(\frac{1}{2}(1)^2 - \frac{1}{2}(0)^2\right) = 4y \left(\frac{1}{2}\right) = 2y$$ The result of the inner integral is 2y.
04

Evaluate the outer integral

Now let's evaluate the outer integral: $$\int_{0}^{2} 2y d y$$
05

Apply the power rule to the outer integral

We'll use the power rule again to integrate with respect to y: $$\int_{0}^{2} 2y d y = 2 \int_{0}^{2} y d y = 2 \left[\frac{1}{2}y^2 \right]_0^2$$
06

Plug in the limits of integration for the outer integral

Finally, we'll use the limits of integration to find the value of the outer integral: $$2 \left[\frac{1}{2}y^2 \right]_0^2 = 2 \left(\frac{1}{2}(2)^2 - \frac{1}{2}(0)^2\right) = 2(2) = 4$$ So the value of the iterated integral is 4.

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Most popular questions from this chapter

Evaluate the following integrals using polar coordinates. Assume \((r, \theta)\) are polar coordinates. A sketch is helpful. $$\int_{R} \frac{d A}{1+x^{2}+y^{2}} ; R=\\{(r, \theta): 1 \leq r \leq 2,0 \leq \theta \leq \pi\\}$$

Find the volume of the following solids.. The solid bounded by the paraboloid \(z=x^{2}+y^{2}\) and the plane \(z=9\).

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Let \(R\) be the unit disk centered at \((0,0) .\) Then $$\iint_{R}\left(x^{2}+y^{2}\right) d A=\int_{0}^{2 \pi} \int_{0}^{1} r^{2} d r d \theta$$ b. The average distance between the points of the hemisphere \(z=\sqrt{4-x^{2}-y^{2}}\) and the origin is 2 (calculus not required). c. The integral \(\int_{0}^{1} \int_{0}^{\sqrt{1-y^{2}}} e^{x^{2}+y^{2}} d x d y\) is easier to evaluate in polar coordinates than in Cartesian coordinates.

Choose the best coordinate system and find the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. The solid inside the cylinder \(r=2 \cos \theta,\) for \(0 \leq z \leq 4-x\).

Areas of circles Use integration to show that the circles \(r=2 a \cos \theta\) and \(r=2 a \sin \theta\) have the same area, which is \(\pi a^{2}\).
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