Switch the order of integration of the double integral: \int_{-10}^{0} \int_{x^2}^{100} f(x,y) dy dx

Question:

Switch the order of integration of the double integral:

{eq}\displaystyle \int_{-10}^{0} \int_{x^2}^{100} f(x,y)\,dy\,dx {/eq}

Double Integrals

To change the order of integration of a double integral like {eq}\displaystyle \int_a^b\int_{u(y)}^{v(y)}f(x,y) dxdy. {/eq}, we will describe the x-simple region of integration {eq}\displaystyle \mathcal{R}=\{(x,y)|\, a\leq y\leq b, u(y)\leq x\leq v(y)\} {/eq} as a y-simple region.

{eq}\displaystyle \displaystyle \mathcal{R}=\{(x,y)|\, c\leq x\leq d, w(x)\leq y\leq h(x)\} {/eq}

And the integral is {eq}\displaystyle \int_c^d\int_{w(x)}^{h(x)}f(x,y) dydx. {/eq}

Answer and Explanation:

To change the order of integration for the integral {eq}\displaystyle \int_{-10}^{0} \int_{x^2}^{100} f(x,y)\,dy\,dx {/eq}, we will first describe the region of integration.

The region of integration is {eq}\displaystyle \mathcal{R}=\{ (x,y)| \ -10\leq x\leq 0, x^2\leq y\leq 100\} {/eq}, which is the region bounded below by the parabola {eq}\displaystyle y=x^2 {/eq} and above by the line {eq}\displaystyle y=100, \text{ for }-10\leq x\leq 0, {/eq} which is an y-simple description.

The points of intersection of the boundary curves of the region are {eq}\displaystyle y=x^2 \text{ with }y=100 \iff (10,100) \text{ and }y=x^2 \text{ with }x=0 \iff (0,0). {/eq}

If we describe the region as an x simple region, we will be able to reverse the order of integration of the original integral.

The region can be described as being bounded on the left by the curve {eq}\displaystyle x=-\sqrt{ y}, {/eq} and on the right by the y-axis {eq}\displaystyle x=0, {/eq} for {eq}\displaystyle 0\leq y\leq 100. {/eq}

Therefore, the region is given as {eq}\displaystyle \mathcal{R}=\{ (x,y)| \ 0\leq y\leq 100, -\sqrt{y}\leq x\leq 0\} {/eq}.

And the integral {eq}\displaystyle \int_{-10}^{0} \int_{x^2}^{100} f(x,y)\,dy\,dx {/eq} with the reverse order of integration is now written as {eq}\displaystyle \boxed{\int_{0}^{100}\int_{-\sqrt{y}}^{0} f(x,y) \space dx\ dy}. {/eq}


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Double Integrals: Applications & Examples

from AP Calculus AB & BC: Help and Review

Chapter 12 / Lesson 14
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