Q. 7 - Calculus

Question

For a four-washer approximation of the volume of the solid obtained from the region between $f\left(x\right)=x^2$ and the$y$-axis between $y=0$ and $y=4$ by rotating around the $x$-axis, illustrate and calculate

(a) $\Delta x$and each $x_{k;}$

(b) some$x_k^{*}$ in each subinterval $\left[x_{k-1},x_k\right]$;

(c) each $f\left(x_k^{*}\right)$;

(d) the volume of the second washer.

Step-by-Step Solution

Answer

a

Step 1: Set up the washer method

For the region between \( f(x) = x^2 \) and the y-axis from \( y = 0 \) to \( y = 4 \), rotated around the x-axis, use four subintervals.

Step 2: Calculate the approximation

Divide \( [0, 4] \) on the y-axis into 4 equal subintervals. For each, the washer has outer radius from the boundary and inner radius from the curve. Sum the volumes \( \pi(R^2 - r^2)\Delta y \).