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Mathematics LibreTexts

Iterated Integrals and Area (Exercises)

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Terms and Concepts

1. When integrating fx(x,y) with respect to x, the constant of integration C is really which: C(x) or C(y)? What does this mean?

Answer:
The constant of integration will be C(y) since y is considered a constant in the integration and thus, the original function f could have terms that are functions of y that were lost when taking the partial derivative with respect to the variable x.

2. Integrating an iterated integral is called _________ __________.

Answer:
iterated integration or multiple integration

3. When evaluating an iterated integral of the form bag2(x)g1(x)dydx, we integrate from _______ to ________, then from _________ to __________.

Answer:
We integrate from bottom to top, then from left to right. Here, from y=g1(x) to y=g2(x), then from x=a to x=b.
More generally, we integrate from curve to curve and then from point to point.

4. One understanding of an iterated integral is that bag2(x)g1(x)dydx gives the _______ of a plane region.

Answer:
area

Problems

In Exercises 5-10, evaluate the integral and subsequent iterated integral.

5. (a) 52(6x2+4xy3y2)dy
(b) 2352(6x2+4xy3y2)dydx

Answer:
(a) 52(6x2+4xy3y2)dy =(6x2y+2xy2y3)|52 =30x2+50x125(12x2+8x8) =18x2+42x117
(b) 2352(6x2+4xy3y2)dydx =23(18x2+42x117)dx =(6x3+21x2117x)|23 =48+84234(162+189+351) =102378 =480

6. (a) π0(2xcosy+sinx)dx
(b) π/20π0(2xcosy+sinx)dxdy

7. (a) x1(x2yy+2)dy
(b) 20x1(x2yy+2)dydx

Answer:
(a) x1(x2yy+2)dy =(x2y22y22+2y)|x1 =x42x22+2x(x2212+2) =x42x2+2x32
(b) 20x1(x2yy+2)dydx =20(x42x2+2x32)dx =(x510x33+x232x)|20 =321083+430 =96308030+3030 =2315

8. (a) y2y(xy)dx
(b) 11y2y(xy)dxdy

9. (a) y0(cosxsiny)dx
(b) π0y0(cosxsiny)dxdy

Answer:
(a) y0(cosxsiny)dx =(sinxsiny)|y0 =sin2y
(b) π0y0(cosxsiny)dxdy =π0sin2ydy =π01cos2y2dy =y12sin2y2|π0 =π20 =π2

10. (a) x0(11+x2)dy
(b) 21x0(11+x2)dydx

Answer:
(a) x0(11+x2)dy =y1+x2|x0 =x1+x2 
(b) 21x0(11+x2)dydx =21(x1+x2)dx =12ln(1+x2)|21 =12ln(5)12ln(2) =12ln(52)

In Exercises 11-16, a graph of a planar region R is given. Give the iterated integrals, with both orders of integration dydx and dxdy, that give the area of R. Evaluate one of the iterated integrals to find the area.

11.
13111.PNG

Answer:
Area=41121dydx =12411dxdy
12411dxdy =12x|41dy =123dy =3y|12 =3(3) =9units2

12.
13112.PNG

13.
13113.PNG

Answer:
Area=427xx11dydx =31y+121dxdy+537y21dxdy
427xx11dydx =42y|7xx1dx =42(82x)dx =(8xx2)|42=321616+4 =4units2

14.
13114.PNG

15.
13115.PNG

Answer:
Area=10xx41dydx =104yy21dxdy
10xx4dydx =10y|xx4dx =10(xx4)dx =(23x3/2x55)|10 =2315 =1015315 =715units2

16.
13116.PNG

In Exercises 17-22, iterated integrals are given that compute the area of a region R in the xy-plane. Sketch the region R, and give the iterated integral(s) that give the area of R with the opposite order of integration.

17. 224x20dydx

Answer:
224x20dydx =404y4ydxdy
Figure14-1-Ex19a.pngFigure14-1-Ex19b.png

18. 1055x255xdydx

19. 2224y20dxdy

Answer:
2224y20dxdy =401216x21216x2dydx
Fig14-1-Ex19a.pngFig14-1-Ex19b.png

20. 339x29x2dydx

21. 10yydxdy+41yy2dxdy

Answer:
10yydxdy+41yy2dxdy =21x+2x2dydx
Figure14-1-Ex21a.pngFigure14-1-Ex21b.png

22. 11(1x)/2(x1)/2dydx


Iterated Integrals and Area (Exercises) is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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