Use a triple integral to find the volume of the given solid. The solid bounded by the parabolic cylinder y = x2 and the planes z = 0, z = 4, y = 16.

Answer:

The answer is "[tex]\bold{\frac{1024}{3}}[/tex]".

Step-by-step explanation:

[tex]= \int\limits^{4}_{-4} \int\limits^{4}_{0} \int\limits^{16}_{x^2} dy\ dx \ dz \\\\= 2 \int\limits^{4}_{0} \int\limits^{4}_{0} \int\limits^{16}_{x^2} dy\ dx \ dz \\\\ = 2 \int\limits^{4}_{0} \int\limits^{4}_{0} (16-x^2) dx \ dz \\\\= 2 \int\limits^{4}_{0} \int\limits^{4}_{0} 16x -\frac{x^3}{3} dx \ dz \\\\= 2 (16x -\frac{x^3}{3})^4_{0} \ dz \\\\= 2 (4) (16x -\frac{64}{3}) \\\\=8(\frac{2}{3} \times 64) \\\\=(\frac{16\times 64}{3} ) \\\\= \frac{1024}{3}[/tex]

The volume of the given solid is [tex]\frac{1024}{3}[/tex] cubic units.

In this question we must use the triple integral formula in rectangular coordinates to determine the volume of the solid ([tex]V[/tex]):

[tex]V = \iiint dy\,dx\,dz[/tex] (1)

The solid is constrained by the following intervals:

[tex]y \in [x^{2}, 16][/tex], [tex]x \in [-4, 4][/tex], [tex]z\in [0, 4][/tex]

Then, the triple integral is now described:

[tex]V = \int\limits_{0}^{4}\int\limits_{-4}^{4}\int\limits_{x^{2}}^{16} dy\,dx\,dz[/tex]

Now we proceed to integrate thrice to obtain the volume:

[tex]V = \int \limits_{0}^{4}\int \limits_{-4}^{4} (16-x^{2})\,dx\,dz[/tex]

[tex]V = \int\limits_{0}^{4} \left(16\cdot x - \frac{x^{3}}{3}\right)\left|\limits_{-4}^{4} dz[/tex]

[tex]V = \frac{256}{3} \int\limits_{0}^{4} dz = \frac{1024}{3}[/tex]

The volume of the given solid is [tex]\frac{1024}{3}[/tex] cubic units.

We kindly invite to check this question on triple integrals: https://brainly.com/question/19484372

if there are 5 fish and 1 of the dies how many are left ​

55 points and Brainliest!

A taxi company charges $3 per ride plus $1.50 for each mile traveled. Andrea wants to spend no more than $24 on a taxi ride. How many miles can Andrea travel? Select all that apply.

a package of 4 pairs of isulated gloves costs $33.96. what is the unit price of gloves​

Is anyone willing to answer this question I can’t do it

The mapping diagram shows a functional relationship.

A mapping diagram shows a relation, using arrows, between domain and range for the following ordered pairs: (8, 4), (negative two-thirds, 3), (11, negative 1), (4, one-half).

Complete the statements.

Consider a 25 factorial design. (a) How many factors and levels are considered in this factorial experiment

HELP!!!!!!!! I LOVE YOU AND ILL EVEN GIVE BRAIN

The average cost of a compact car at a certain dealership over the last 10 years has ranged from $15,000 to $24,500. Suppose you were going to plot these points on a coordinate grid. (a) What is a good scale [range] would be good to use for the y-axis to make a nicely sized graph? Explain your reasoning. (b) What is a good interval [step size] would be good to use for the y-axis to make a nicely sized graph? Explain your reasoning. (c) What is a good scale [range] would be good to use for the x-axis to make a nicely sized graph? Explain your reasoning. (d) What is a good interval [step size] would be good to use for the x-axis to make a nicely sized graph? Explain your reasoning. Note: The step size the distance between the marks on the graph, and the range is the maximum and minimum values (ex: -10 to 10).

Bart drives 45 miles in 30 minutes. If he drove 3 hours in total at the same rate how far did he go! How many minutes did Bart drive in total!

An angle measures 50° less than the measure of its supplementary angle. What is the measure of each angle?

I mark as brainliest ​

Question 1 Part A: How many beads did Jiao send out? *
Wooden Beads
Oak
4,525
Maple
6,950
Ash
3,720
Ebony
2,475
Bayong
1,250

A stack of books is 62 inches high. How high is it in feet and inches?

Can someone plz plz help me with the last 2 problems #5 and #6

A one dimensional object with no endpoint is a?
A.ray
B.line segment
C.line
D.point

Graph the system of linear inequalities on the coordinate plane.
y [tex]\geq[/tex] 1/2x + 2 1/2
y< 1/5 x + 6
a. Shade the solution to the system of inequalities.
b. Pick a point and show your solution is correct.

What is 5 8/100
as a decimal?
O
A. 5.008
0
B. 5.08
O
C. 5.8
0 D. 58

If the volleyball team washes 10 more cars, then they will have met their overall goal of
washing 60 cars to raise money for their season. Let c represent the number of cars that
the volleyball team has washed so far. Which of the following equations can be used to
solve for c?

What is an equation of the line that passes through the points (-3,-5) and
(-3,-8)?

The length of a rectangle is 10 meters more than the width. If 3 meters is added to both the length and the width, the new area will be 299 square meters. Find the length and width of the original rectangle.
HELP HELP HELP HELP

Matt was standing 3 feet from the edge of the pool he walked away from the pool for four seconds then he was 12 feet away from the edge of the pool

Kyle invested $ 500 and received $ 650 after three years. What had been the interest rate?

A researcher is exploring the notion that there is an economy of scale in raising children. While having one child might add to a family's financial stress, the second or third one might not be as "expensive" since they would already be sharing one roof, could share clothing, etc. The researcher has access to a large community of parents where 15% have one child under 18 in their home, 65% of parents have two children, and 20% of parents have three or more children. Rounded to the nearest whole number, the probability that any given family who is selected has two or more children at home is ________ %?

pleas help me ill give brainleyest

A giant garter snake is at least 64 inches long. If 1 inch is equal to
2.5 cm, how long is the Giant garter snake in cm?

The US emerged as a world power because of its success in
A. the Korean War
B. World War I
C. World War II
D. the Cold War

3x+15=24 what is the x equal too

Given parallelogram ABCD, the measure of angle C

identify the first term of the algebraic expression. Indicate whether the term is a variable or a constant term.
5y + 8b