Question

Use the general slicing method to find the volume of the following solid. The solid whose base is the region bounded by the curve y=38cosx and the​ x-axis on − π 2, π 2​, and whose cross sections through the solid perpendicular to the​ x-axis are isosceles right triangles with a horizontal leg in the​ xy-plane and a vertical leg above the​ x-axis. A coordinate system has an unlabeled x-axis and an unlabeled y-axis. A curve on the x y-plane labeled y equals 38 StartRoot cosine x EndRoot starts on the negative x-axis, rises at a decreasing rate to the positive y-axis, and falls at an increasing rate to the positive x-axis. The region below the curve and above the x-axis is shaded. A right triangle extends from the x y-plane, where one leg is on the x y-plane from the x-axis to the curve and is perpendicular to the x-axis, and the second leg is above the x-axis and is perpendicular to the x y-plane. y=38cosx

Answer 1

Answer:

The volume of the solid = 1444

Step-by-step explanation:

Given that:

The region of the solid is bounded by the curves $y = 38 \sqrt{cos \ x}$ and the axis on $[-\dfrac{\pi}{2}, \dfrac{\pi}{2}]$

using the slicing method

Let say the solid object extends from a to b and the cross-section of the solid perpendicular to the x-axis has an area expressed by function A.

Then, the volume of the solid is ;

$V = \int ^b_a \ A(x) \ dx$

However, each perpendicular slice is an isosceles leg on the xy-plane and vertical leg above the x-axis

Then, the area of the perpendicular slice at a point $x \ \epsilon \ [-\dfrac{\pi}{2},\dfrac{\pi}{2}]$ is:

$A(x) =\dfrac{1}{2} \times b \times h$

$A(x) =\dfrac{1}{2} \times(38 \sqrt{cos \ x})^2$

$A(x) =\dfrac{1444}{2} \ cos \ x$

$A(x) =722 \ cos \ x$

Applying the general slicing method ;

$V = \int ^b_a \ A(x) \ dx \\ \\ V = \int ^{\dfrac{\pi}{2} }_{-\dfrac{\pi}{2}} (722 \ cos x) \ dx \\ \\ V = 722 \int ^{\dfrac{\pi}{2}}_{-\dfrac{\pi}{2}} cosx \dx$

$V = 722 [ sin \ x ] ^{\dfrac{\pi}{2}}_{-\dfrac{\pi}{2}}$

$V = 722 [sin \dfrac{\pi}{2} - sin (-\dfrac{\pi}{2})]$

$V = 722 [sin \dfrac{\pi}{2} + sin \dfrac{\pi}{2})]$

$V = 722 [1+1]$

$V = 722 [2]$

V = 1444

∴ The volume of the solid = 1444