Q. 6 - Calculus

Question

For a four-disk approximation of the volume of the solid obtained from the region between$f\left(x\right)=5-x^2$ and the$x$-axis between $x=0$ and $x=4$ 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 disk.

Step-by-Step Solution

Answer

A

Step 1 : Introduction Part (a)

A

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.