A frustum of a sphere is made by cutting a sp | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The equation of the circle representing the cross-section of the sphere is $x^2 + y^2 = 5^2 = 25$.The curved surface area $S$ of a solid of revolu

Vedclass · Accepted Answer

$10\pi \, cm^2$

Vedclass · Answer

(B) The equation of the circle representing the cross-section of the sphere is $x^2 + y^2 = 5^2 = 25$.The curved surface area $S$ of a solid of revolution generated by rotating the arc of the circle around the $x$-axis is given by $S = 2\pi \int_{x_1}^{x_2} y \, ds$,where $ds = \sqrt{1 + (\frac{dy}{dx})^2} \, dx$.From $x^2 + y^2 = 25$,we have $2x + 2y \frac{dy}{dx} = 0$,so $\frac{dy}{dx} = -\frac{x}{y}$.Then $ds = \sqrt{1 + (-\frac{x}{y})^2} \, dx = \sqrt{\frac{y^2 + x^2}{y^2}} \, dx = \sqrt{\frac{25}{y^2}} \, dx = \frac{5}{y} \, dx$.The limits of integration are from $x_1 = 2 \, cm$ to $x_2 = 2 + 1 = 3 \, cm$.Thus,$S = 2\pi \int_{2}^{3} y \cdot \frac{5}{y} \, dx = 2\pi \int_{2}^{3} 5 \, dx$.$S = 2\pi [5x]_{2}^{3} = 2\pi (15 - 10) = 10\pi \, cm^2$.

$A$ frustum of a sphere is made by cutting a sphere with two parallel planes. If the radius of the sphere is $5 \, cm$ and the distance between the planes is $1 \, cm$,what is the curved surface area of the frustum when the distance of the first plane from the center of the sphere is $2 \, cm$?

Similar Questions

The area of the region bounded by the parabola $y = x^2 + 2$,the $X$-axis,and the lines $x = 1$ and $x = 2$ is . . . . . . sq. units.

The area of the region bounded by the curve $y = \cos x$ between $x = 0$ and $x = \pi$ is

The line $x=\frac{\pi}{4}$ divides the area of the region bounded by $y=\sin x$,$y=\cos x$ and the $x$-axis $\left(0 \leq x \leq \frac{\pi}{2}\right)$ into two regions of areas $A_1$ and $A_2$. Then $A_1 : A_2$ equals (in $: 1$)

The area bounded by the parabola $y = 4x^2$,the $y$-axis,and the lines $y = 1$ and $y = 4$ is:

The area of the smaller part between the circle $x^2+y^2=4$ and the line $x=1$ is . . . . . . sq. units.

Vedclass Test Series

Exam Paper Generator

Online Exam Module