(a) find the area of r. (b) region r is the base of a solid. for the solid, each cross section perpendicular to the x-axis is a square. write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid. (c) the horizontal line y k divides r into two regions of equal area. write, but do not solve, an equation involving one or more integrals whose solution gives the value of k

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JanaMiqdad8062 JanaMiqdad8062

15-12-2022
Mathematics

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(a) find the area of r. (b) region r is the base of a solid. for the solid, each cross section perpendicular to the x-axis is a square. write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid. (c) the horizontal line y k divides r into two regions of equal area. write, but do not solve, an equation involving one or more integrals whose solution gives the value of k

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abhinavjh321 abhinavjh321 · Answer 1 · 2022-12-21T07:35:01+01:00

The area of region R is given by the integral ∫[0,5] (x^2+2) dx = 41.6667,The volume of the solid is given by the integral ∫[0,5] (x^2+2)^2 dx and the equation to find the value of k is given by the equation ∫[0,k] (x^2+2) dx = ∫[k,5] (x^2+2) dx.

a) We must integrate the function (x2+2) over the range [0,5] in order to determine the area of region R. The ensuing integral provides this information:

∫[0,5] (x^2+2) dx = 41.6667

b) To get the solid's volume, we must integrate the function (x2+2)2 over the range [0,5]. The ensuing integral provides this information:

∫[0,5] (x^2+2)^2 dx \sc) We must construct an equation that asserts that the area of the region below the line y=k is equal to the area of the region above the line y=k in order to get the value of k. The following equation yields this:

∫[0,k] (x^2+2) dx = (x2 + x2)[k,5] dx

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