The base of a solid is the region in the first quadrant between the graph of y=x2 and the x -axis for 0≤x≤1 . For the solid, each cross section perpendicular to the x -axis is a semicircle. What is the volume of the solid?

Answer:

The volume of the solid is π/40 cubic units.

Step-by-step explanation:

Please refer to the graph below.

Recall that the area of a semi-circle is given by:

[tex]\displaystyle A=\frac{1}{2}\pi r^2[/tex]

The volume of the solid will be the integral from x = 0 to x = 1 of area A. Since the diameter is given by y, then the radius is y/2. Hence, the volume of the solid is:

[tex]\displaystyle V=\int_0^1\frac{1}{2}\pi \left(\frac{y}{2}\right)^2\, dx[/tex]

Substitute:

[tex]\displaystyle V=\frac{1}{2}\pi\int_0^1\left(\frac{x^2}{2}\right)^2\, dx[/tex]

Simplify:

[tex]\displaystyle V=\frac{1}{2}\pi \int_0^1\frac{x^4}{4}\, dx[/tex]

Integrate:

[tex]\displaystyle V=\frac{1}{2}\pi \left[\frac{x^5}{20}\Big|_0^1\right][/tex]

Evaluate:

[tex]\displaystyle V=\frac{\pi}{40}\left((1)^5-\left(0\right)^5\right)=\frac{\pi}{40}\text{ units}^3[/tex]

The volume of the solid is π/40 cubic units.

Volume of a solid is the measure of the 3 dimensional space it occupies. The volume of the considered solid is obtained as [tex]\dfrac{\pi}{40} \: \rm unit^3[/tex]

For that, we can try to find infinitesimally small 3-d region's volume, and then integrate that region over the dimensions available to get the total volume of the specified region.

We can also use the fact that continuous curves are almost linear and non-changing in infinitely zoomed region.

The given solid has base bounded by x-axis, [tex]y=x^2[/tex] and 0≤x≤1

Its three dimensional region is along the z axis, for each x, there is a semicircle perpendicular with radius being 'y'.

If we take [tex]dx[/tex]x-axis, then the curve  [tex]y=x^2[/tex]cylinder(split from height because of semicircle)) with diameter y, and height [tex]dx[/tex]volume is : [tex]V_{dx} = \dfrac{1}{2} \times \pi (\dfrac{y}{2})^2 \times dx = \dfrac{\pi (x^2)^2}{8}dx = \dfrac{\pi x^4}{8} dx[/tex]

Integrating this for 0≤x≤1, we will get the volume of the three dimensional region needed as:

[tex]V = \int_0^4V_{dx} = \int_0^1 \dfrac{\pi x^4}{8} dx = \dfrac{\pi}{8} [\dfrac{x^5}{5}]^1_0 = \dfrac{\pi 1^4}{40} = \dfrac{\pi}{40}[/tex] (in cubic units).

Thus, the volume of the considered solid is obtained as [tex]\dfrac{\pi}{40} \: \rm unit^3[/tex]

Learn more about volume of three dimensional region here:

https://brainly.com/question/338504

Jonathan went shopping for a new pair of pants. Sales tax where he lives is 4.75%. The price of the pair of pants is $20. Find the total price including tax. Round to the nearest cent.

Janie ordered boxed lunches for a student advisory committee meeting. Each lunch costs $4.25. The total cost of the lunches is $53.75, including a $7 delivery fee.

Part A

Which equation will help you find x, the number of boxed lunches Janie ordered?
a 4.25x = 60.75
b 4.25 + 7x = 53.75
c 4.25x + 7 = 53.75
d 4.25(x + 7) = 53.75

Part B

Solve the equation from Part A to find x.

x = _____

What is the volume of a sphere with the diameter = 10 cm and the height = 200 cm ​

The radius of a circle is changing at the rate of 1/π inches per second. At what rate, in square inches per second, is the circle’s area changing when the radius is 5 inches?

Pls help !!!! Due today

if i have 2 pans of brownies and i give away 1 5/6. How many browniea do i have left? in a fraction?
​

factoring review 2 terms 1-8

Find the measurement of the angle labeled X


PLEASE IM REALLY BAD AT THESE QUICK!!!!!!!!!!!!!

Evaluate the expression if y = -4. | 2/y + 1/2 (y+6)

DO NOT SEND ANY LINKS.

What does MIJ equal ?

What is the volume, in cubic centimeters, of a cube with an edge length of 15 centimeters?

In the figure below, what is the measure of _x?
30
х
O A. 70°
OB. 300
o C. 60°
D. 90°

Solve for x
15 points for prize

Can someone help me please?

The diameter of each circular hole in a paint palette is 42 millimeters.
Find the circumference of each hole. Use 3.14 as an approximation for .
A 128 millimeters
B 131.88 millimeters
C 120 millimeters
D 42 millimeters

evaluate: (_1_i)^(5/3)​

Find l.
10 in.
l
l = [?] ir
1
1
6 in.

Question 1 of 10
What is the solution to this equation?
9x - 4(x - 2) = x + 20
A. X = 3
B. X = 7
C. X = -7
D. X = -3

Find the quotient of 376 divided by 4

Please help givin brainliest

The graph of a linear function is shown on the coordinate grid.
What is they y-intercept of the graph of the linear function?

There are 20 pieces of fruit in a bowl and 5 of them are apples. What percent of the fruit are apples?

I need help on homework its due tomorrow

What slope of the points (9,7) and (3,3)

HELP ASAP!! I will give brainlist!! THANK YOU!

What is Landscape evolution theory
​

85% of x is 51. Write an equation that represents the situation.

Line u has a slope of Line v is parallel to line u. What is the slope of line v? v יל Simplify your answer and write it as a proper fraction, improper fraction, or integer.​

Find the area of the composite figure?

how many square feet is in a acre