Answer to Question #153668 in Calculus for james

Question #153668

f(x)=6sin^(-1)\sqrt(1-x^(2))


1
Expert's answer
2021-01-12T13:11:34-0500

"f(x)=6sin^{-1}(\\sqrt{1-x^2})\\\\\n~~~~~~~~~= \\begin{cases}\n 6cos^{-1}(x) &\\text{if }-1\\leq x\\leq0 \\\\\n 6\\pi-6cos^{-1}(x) &\\text{if }\\;0\\leq x\\leq1\n\\end{cases}\\\\\\;\\\\\n\n\\text{Domain of }f(x) \\;\\text{is}: x\\in [-1,1]\\\\\\;\\\\\n\\text{Range of }f(x)\\;\\text{is} : f(x)\\in \\left[0,3\\pi\\right]\\\\\\;\\\\\n\nf'(x)= \\begin{cases}\n -\\dfrac{3}{\\sqrt{1-x^2}}<0 &\\text{if }-1\\leq x<0 \\\\\n +\\dfrac{3}{\\sqrt{1-x^2}}>0 &\\text{if }\\;0< x\\leq1\n\\end{cases}"


"f(x) \\text{ is not differentiable at } x=0"




"\\text{Now, }\\\\\n{\\displaystyle\\int}cos^{-1}\\left(x\\right)\\,\\mathrm{d}x\n=xcos^{-1}\\left(x\\right)-\\sqrt{1-x^2}+C\\\\\n\\text{Area bounded by the curve and the x-axis is :-}\\\\\n{{\\displaystyle\\int}_{1}^{-1}{f(x)}\\,\\mathrm{d}x =2\\times {\\displaystyle\\int}^1_{0}{f(x)}\\,\\mathrm{d}x=2\\times6=12}"


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