I need the answer ASAP plz. Ty!!! ---### Volume of a Solid Generated by Cross SectionsLet the region \( R \) be the area enclosed by the function \( f(x) = x^3 + 1 \), the horizontal line \( y = 0 \) and the vertical lines \( x = 0 \) and \( x = 2 \). If the region \( R \) is the base of a solid such that each cross-section perpendicular to the \( x \)-axis is a rectangle whose height is twice the length of its base in the region \( R \), find the volume of the solid. You may use a calculator and round to the nearest thousandth.#### Graph Interpretation:The graph included depicts the function \( f(x) = x^3 + 1 \). It shows the following details:- The horizontal axis (x-axis) ranges from approximately \( -2 \) to \( 3 \).- The vertical axis (y-axis) ranges from \( 0 \) to \( 15 \).- The function starts increasing steeply after passing the x-axis and continues to rise rapidly.- The vertical lines \( x = 0 \) and \( x = 2 \) mark the boundaries on the x-axis for the region \( R \).### Finding the Volume:To find the volume of the solid formed, we need to calculate the integral of the area of the cross-sections perpendicular to the x-axis from \( x = 0 \) to \( x = 2 \).Given the height of each rectangle is twice the base, we start by noting the base of each rectangle at a point \( x \) is \( y = f(x) \). Hence, the length of the base is \( f(x) \) and the height is \( 2f(x) \).The area \( A(x) \) of each rectangular cross-section as a function of \( x \) is:\[ A(x) = \text{base} \times \text{height} = f(x) \times 2f(x) = 2(f(x))^2 = 2(x^3 + 1)^2 \]The volume \( V \) of the solid is then the integral of \( A(x) \) from \( x = 0 \) to \( x = 2 \):\[ V = \int_{0}^{2} 2(x^3 + 1)^2 \, dx \]

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Let the region R be the area enclosed by the function f(x) = x +1, the horizontal line y = 0 and the vertical lines x = solid such that each cross section perpendicular to the x-axis is a rectangle whose height is twice the length of its base in the region R, find the volume of the solid. You may use a calculator and round to the nearest thousandth. || O and x = 2. If the region R is the base of a 15 14 13 12 11 10 9 6. 4 2.

Let the region R be the area enclosed by the function f(x) = x +1, the horizontal line y = 0 and the vertical lines x = solid such that each cross section perpendicular to the x-axis is a rectangle whose height is twice the length of its base in the region R, find the volume of the solid. You may use a calculator and round to the nearest thousandth. || O and x = 2. If the region R is the base of a 15 14 13 12 11 10 9 6. 4 2.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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