. Differential and integral calculus, an introductory course for colleges and engineering schools. C/ = - fVa2-z2efc = a& f2cos20d0 = ^[0 + sin0cos0l Hence a& 7ra6 The entire area of the ellipse is therefore last integration shows that when we substitute a new variable IN PLACE OF THE ARGUMENT IN THE INTEGRAND OF A DEFINITE INTEGRAL, WE MUST TAKE CARE TO MAKE PROPERCHANGES IN THE LIMITS OF THE IN-TEGRAL. Example 4. Let us find the areaincluded between the parabolas y2 = 2mx and x2 = 2my. From the figure V = OMBNO = 0MBAO - Uf. 2mx^dx ^«Mr-&M 1m0 The coordinates of 0 and B, the

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. Differential and integral calculus, an introductory course for colleges and engineering schools. C/ = - fVa2-z2efc = a& f2cos20d0 = ^[0 + sin0cos0l Hence a& 7ra6 The entire area of the ellipse is therefore last integration shows that when we substitute a new variable IN PLACE OF THE ARGUMENT IN THE INTEGRAND OF A DEFINITE INTEGRAL, WE MUST TAKE CARE TO MAKE PROPERCHANGES IN THE LIMITS OF THE IN-TEGRAL. Example 4. Let us find the areaincluded between the parabolas y2 = 2mx and x2 = 2my. From the figure V = OMBNO = 0MBAO - Uf. 2mx^dx ^«Mr-&M 1m0 The coordinates of 0 and B, the intersections of the curves, are (0, 0) and(2 m, 2 m). Hence „2to X2 2 /— -—dx = -0 -0 2m 3 = 2(2m)2 _ (2m)2 (2m)2 = 4m_2 3 3 ~ 3 3 It appears also that ONBAO = U. By interchanging x and y in formula (3), we have f xdy (3) U 230 INTEGRAL CALCULUS 161 as the formula which gives the area bounded by the curve x = f(y), theaxis of y and the abscissas of the points whose ordinates are a and b. Example 5. The curve in the figureis the cubical parabola y = x*. Thenx = y% and by formula (3)

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