The area of R is equal to the area of S when the integral of y=x from 0 to b is equal to the integral of y=cos(x) from 0 to b. To solve for b, we can set the two integrals equal to each other and solve for b.
Integral of y=x from 0 to b = Integral of y=cos(x) from 0 to b
∫x dx from 0 to b = ∫cos(x) dx from 0 to b
(1/2)x^2 from 0 to b = sin(x) from 0 to b
(1/2)b^2 = sin(b) - sin(0)
(1/2)b^2 = sin(b)
b^2 = 2sin(b)
b = sqrt(2sin(b))
We can solve for b numerically using a calculator or computer program. The answer is b = 1.404.
R is the region below the curve y=x and above the z-axis from x=0 to x=b, where b is a positive constant. S is the region below the curve y=cos(x) and above the x-axis from x=0 to x=b. For what value of b is the area of R equal to the area of S?
I know the answer is 1.404, but I need an explanation as to why. Please and thank you.
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