Engineering Mathematics – Multiple Integrals |Practice Questions on Double & Triple Integrals, Change of Order, and Coordinate Transformations

1. Sketch the region of integration of the following double integrals and evaluate them:
4 x2 2  a 1cos 

 x 
1 x a
(i)  2
 y dydx
2
(ii)   e y / x dydx (iii)   rdrd (iv)  r
3
sin cos drd .
0 x 0 0 0 a sin 0 0

2. Evaluate the following double integrals over the given region R:

 x 
 y 2 dxdy ; R is the region bounded by y  x and y 2  4 x
2
(i)
R

x2 y2
 x ydxdy ; R is the region enclosed by the ellipse   1 in the first quadrant.
3
(ii)
R a2 b2

rdrd
(iii)  ; R is the region over one loop of the lemniscate r 2  a 2 cos 2 .
R a2  r 2

 r sindrd  2a ; R is the semi-circle r  2a cos  above the initial line.
2
(iv)
R

3. Change the order of integration in the following integrals and hence evaluate them:
4 y 2 x 2 a a
y2
3 4 a 2 ax 1
x
(i)   ( x  y)dxdy (ii)   xydydx (iii)   x2  y2
dydx (iv)   dydx .
0 1 0 x2 0 x 0 ax y4  a2 x2
4a

4. Using double integrals, find the following areas given by A:
(i) A is bounded by the parabolas y 2  4  x and y 2  x .

(ii) A is the region bounded by the curve x( y 2  a 2 )  a3 and its asymptote.

(iii) A is the larger of the two areas into which the circle x 2  y 2  64a 2 is divided by the

parabola y 2  12ax .

(iv) A is the area lying inside the circle r  a sin and outside the cardioid
r  a1  cos   .



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