Let the region mathrm{R} be the area enclosed by the function f(x)=3 x^{2} and g(x)=4 x. If the region mathrm{R} is the base of a solid such that each cross section perpendicular to the x axis is an isosceles right triangle with a leg in the region mathrm{R}, find tha volume of the solid. You may use a calculator and round to the nearest thousandth.

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Accepted Answer

Step:1 ：Find the points of intersection between (f(x) = 3x^2) and (g(x) = 4x) by setting (f(x) = g(x))Step:2 ：Integrate the area of the cross-sectional triangles along the x-axis from the left intersection point to the right intersection point using the formula (A(x) = frac{1}{2}(-3x^2 + 4x)^2)Step:3 ：Evaluate the integral to find the volume of the solid: (boxed{0.632}) cubic units

Answer

Step: 1 ：Find the points of intersection between (f(x) = 3x^2) and (g(x) = 4x) by setting (f(x) = g(x))Step: 2 ：Integrate the area of the cross-sectional triangles along the x-axis from the left intersection point to the right intersection point using the formula (A(x) = frac{1}{2}(-3x^2 + 4x)^2)Step: 3 ：Evaluate the integral to find the volume of the solid: (boxed{0.632}) cubic units

Answer

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