Calculus

Let R be the region enclosed by the graphs y=e^x, y=x^3, and the y axis.

A.) find R
B.) find the volume of the solid with base on region R and cross section perpendicular to the x axis. The cross sections are triangles with height equal to 3 times the length of their base.

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  1. the first place where e^x = x^3 is near x = 1.857

    So, we have a narrow strip extending from x=0 out to about (1.857,6.406)

    The solid is a set of triangles of base e^x-x^3 and height 3 times that.

    So, the volume of the solid is

    v = ∫[0,1.857] (1/2)(e^x-x^3)* (3)(e^x-x^3) dx
    = ∫[0,1.857] 3/2 (e^x-x^3)^2 dx
    = 5.267

    You have to expand the polynomial and use integration by parts to integrate x^3 e^x

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