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 bounded by the curve $4y^{2} = x^{2}(4-x)(x-2)$ is equal to ...... .

The area bounded by the curve $y=x|x|$,the $x$-axis,and the ordinates $x=-1$ and $x=1$ is given by [Hint: $y=x^{2}$ if $x>0$ and $y=-x^{2}$ if $x < 0$].

The area of the region bounded by the curves $y = |x - 4|$, $x = 3$, $x = 5$, and the $X$-axis is

Area of the region (in sq. units) bounded by the curve $y=x^2-5x+4$,$x=0$,$x=2$ and the $X$-axis is

The area of the region bounded by the lines $y=mx$,$x=1$,$x=2$,and the $x$-axis is $6$ sq. units. Then,the value of $m$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module