Answer: Choice C
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Explanation:
The graph is shown below. The base of the 3D solid is the blue region. It spans from x = 0 to x = 1. It's also above the x axis, and below the curve 
Think of the blue region as the floor of this weirdly shaped 3D room.
We're told that the cross sections are perpendicular to the x axis and each cross section is a square. The side length of each square is 
where 0 < x < 1
Let's compute the area of each general cross section.
We'll be integrating infinitely many of these infinitely thin square slabs to find the volume of the 3D shape. Think of it like stacking concrete blocks together, except the blocks are side by side (instead of on top of each other). Or you can think of it like a row of square books of varying sizes. The books are very very thin.
This is what we want to compute
Apply a u-substitution
u = -2x
du/dx = -2
du = -2dx
dx = du/(-2)
dx = -0.5du
Also, don't forget to change the limits of integration
- If x = 0, then u = -2x = -2(0) = 0
- If x = 1, then u = -2x = -2(1) = -2
This means,
I used the rule that 
which says swapping the limits of integration will have us swap the sign out front.
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Furthermore,
![\displaystyle 0.5\int_{-2}^{0}e^{u}du = \frac{1}{2}\left[e^u+C\right]_{-2}^{0}\\\\\\= \frac{1}{2}\left[(e^0+C)-(e^{-2}+C)\right]\\\\\\= \frac{1}{2}\left[1 - \frac{1}{e^2}\right]](https://tex.z-dn.net/?f=%5Cdisplaystyle%200.5%5Cint_%7B-2%7D%5E%7B0%7De%5E%7Bu%7Ddu%20%3D%20%5Cfrac%7B1%7D%7B2%7D%5Cleft%5Be%5Eu%2BC%5Cright%5D_%7B-2%7D%5E%7B0%7D%5C%5C%5C%5C%5C%5C%3D%20%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%28e%5E0%2BC%29-%28e%5E%7B-2%7D%2BC%29%5Cright%5D%5C%5C%5C%5C%5C%5C%3D%20%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B1%20-%20%5Cfrac%7B1%7D%7Be%5E2%7D%5Cright%5D)
In short,
![\displaystyle \int_{0}^{1}e^{-2x}dx = \frac{1}{2}\left[1 - \frac{1}{e^2}\right]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_%7B0%7D%5E%7B1%7De%5E%7B-2x%7Ddx%20%3D%20%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B1%20-%20%5Cfrac%7B1%7D%7Be%5E2%7D%5Cright%5D)
This points us to choice C as the final answer.
First, we need to add up all the quantities in ounces. 10 + 14 + 40 = 64. One pound = 16 ounces. 64 / 16 = 4, so the total weight in pounds is 4 lbs.
Answer:
It will take 3.65 seconds at the rate it's being poured.
Step-by-step explanation:
<span>f(x) = -3x^2</span>
insert x=2
y=-3*2^2
y=-3*4
y=-12
so your pair is (2,-12)
A table can be used to find the rate of change by giving you a visual representation of how the operation changes over time for example in the link i'm about to post you can see how much more money is earned as more cars are washed, it is showing how much money is being made OVER TIME, hope this helps :)
now to answer your second question, say you would like to know how many miles Tracy ran over a course of 2 hours, the x axis would be the miles and the y axis would be the time, then you would plot the different points on the graph showing the rate at which she is running
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to answer your last question a linear interval would be a sequence of numbers that when places on a number line they will form a line, a nonlinear interval is a sequence of numbers that don't make a line because there are inequities meaning the numbers in the sequence are all over the place
Hope this helps and i will post the pictures so you can have a visual representaion
