. Applied calculus; principles and applications . mple 3. — Find the area between the parabolaand the circle y^ = 2ax — x^. ax nv2ai-xadxdy^. = 21 iV2ax - x2 - Vo^) dxJo 7ra2 4a^3 Note. — It may be seen that in finding some areas there isno advantage in using double integration, as after the firstintegration with the limits substituted, the remaining in-tegral is what might have been formed at first. There are,however, cases where double integration furnishes the only-method of solution; hence the need for some practice in itsapplication. EXERCISE XXXVI. 1. Find the area between the circle x^
. Applied calculus; principles and applications . mple 3. — Find the area between the parabolaand the circle y^ = 2ax — x^. ax nv2ai-xadxdy^. = 21 iV2ax - x2 - Vo^) dxJo 7ra2 4a^3 Note. — It may be seen that in finding some areas there isno advantage in using double integration, as after the firstintegration with the limits substituted, the remaining in-tegral is what might have been formed at first. There are,however, cases where double integration furnishes the only-method of solution; hence the need for some practice in itsapplication. EXERCISE XXXVI. 1. Find the area between the circle x^ -\- y^ = a^ and the line ya — X. ra /»va2—x2A = I I dxdy = -2 2. Find by double integration the area between the parabolas1/2 = 8 X and x^ = 8y. Ans. 21i 3. Find the area bounded by the circle x^ -\- y^ = 25, the parabolay^ = -^/ x, and the parabola y = ts ^^• Ans. 4. Find by double integration the area of the x^ ifiellipse - + p = 1. 5. Find the area of any right triangle, usingdouble integration. A- f f dxdy = £ {-Ix + aj. dx = — - -7Z +ax \ =7 I h 2 308 INTEGRAL CALCULUS 165. Plane Areas by Double Integration — Polar coor-dinates.— As has been shown in Art. 135(b), the area in polar coordinates of P1OP2,generated by the radiusvector p as ^ increases from$1 to 02 is given by
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