Calculus Area between curves

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If you don't understand the first problem, the following you will be completely lost on the following ones. Here's a picture to help you visualize what's happening.

h(x) in your professor's example is the difference between the two curves. Because you're looking for the area between the two curves, h(x) will be the curve furthest away from the x axis - the curve closest to it.
What my not-to-scale picture is trying to represent the the subtraction of one area from another. If you imagine subtracting the blue area from the purple, you'll be left with the red.

>>290223
Ok So I now see I have to integrate e^-x minus sqrt(x). I also want to assume the intersection of the two equations is my upper bound. SO once I've done that, I guess a calculator is my best friend at that point.

Wait a second, I got another problem where all it's giving me is y=sqrt(49-x^2), x=7 and the axis(es?). It says the axis of rotation is the x axis, I don't get if I have to outline the inner part the y-equation or betweeen x=7 nad the y-equation

>>290243
Could you post the actual question? If it's as simple as you're stating it, you would just rotate y = sqrt(49 - x^2) around the x axis. y = sqrt(49 - x^2) is literally just the upper half of a circle with a radius of 7 so x = 7 is pretty irrelevant. Since this is half a circle being rotate about its midpoint you can be even lazier and just use the volume of a sphere with radius 7.

>>290284
Op here, I finished it and simply found the area of everywhere, BUT inside of y=sqrt(49-x^2) with a bound at x=7. I thought you needed to find the volume of what appears to be a sphere but it looks like we're supposed to find the volume of inbetween y-equation and x=7. If a problem comes up that I have 0 leads on I'll be sure to brb.

OP here with more depression. I thought I knew what this problem was asking until I re-read it 5 times and I'm confused. The problem reads exactly "The region in the first quadrant bounded by the line y=7, below by the curve y=sqrt(7x), and on the left by the y-axis, about the line y=7." What am I reading?? Skipping this problem into I get some help.

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>>290379
It's always helpful to graph the curves if you don't know what you're integrating. They want the area below y = 7 and above y = sqrt(7x). The y-axis is your starting bound and the intersection of the two curves is your right ending bound.

>>290412
That's what I originally thought I had to integrate, yet as I kept reading the question I thought I had to integrate infinite area which is dumb, but I still believed it. I started working on this at 5 pm with limited breaks and am only on #12, I need to learn to be more efficient and stop letting time run away. Thanks for the help!