Use trigonometric substitution to verify that

$$\int\limits_{0}^{x}\sqrt{a^{2}-t^{2}}dt$$〖√(a^2-t^2 ) dt〗=1/2 a^2 sin^(-1)⁡〖x/a〗+1/2 x√(a^2-x^2 )

Use the figure to give trigonometric interpretations of both terms on the right side of the equation in part (a).

Question

Use trigonometric substitution to verify that

$$\int\limits_{0}^{x}\sqrt{a^{2}-t^{2}}dt$$〖√(a^2-t^2 ) dt〗=1/2 a^2  sin^(-1)⁡〖x/a〗+1/2 x√(a^2-x^2 )

Use the figure to give trigonometric interpretations of both terms on the right side of the equation in part (a).

roadjester · Answer

$$\int\limits\limits_{0}^{x}\sqrt{a^{2}-t^{2}}dt=(1/2)a^2\sin^{-1} (x/a)+(1/2)x \sqrt{a^2-x^2}$$

roadjester · Answer

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