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Problem 30 Hard Difficulty

Find the volume under the given surface $z=f(x, y)$ and above the rectangle with the given boundaries.
$$z=3 x+10 y+20 ; \quad 0 \leq x \leq 3,-2 \leq y \leq 1$$




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Video Transcript

because here we want to find the volume bounded below this function. Seeing equals three x plus can why? Plus twenty And the region underneath the graph. And that we want to find the volume above Sierra three. Cross negative two one. But all this to say, we just want this double integral. So it's not gonna matter whether we do extra life first. It's just a rectangle. So we can do sirah to three on X and two to one on Why three x Then why? I was twenty. And we have the Y on the inner the excellent outer. Okay, so now I just need to find some anti derivatives. Clear to three. This is going to be three X. Why pull us five? Why squared? It was twenty. Why? Evaluated from negative to one. Okay. And so we can just value each one of these Live I squared. Why? From one minus two. Okay, so why evaluated from Orinda minus two is just going to give us three. Just told it and that in a bomb. So let's going to be nine x and then plus five. Okay, here, we have to do one squared minus negative too squared and then plus twenty again just times The difference between those two, which is three said twenty times three is going to be sixty. Okay, so we'Ll simplify things down a little bit. We have my next and then this is going to be one My sport threesome minus fifteen. The sixty was forty five t x. Okay, so that's running appeared. And then we just need an ex anti derivatives that'LL be nine has Ex Cleared plus forty five X that evaluated from zero to three course zero It'LL just be easier assholes throwing three. So this is eighty one over to you, plus forty five times three. So that should be one thirty five. And now one thirty five is to seventy over two to seventy plus eighty one will give us three. Fifty one Vicky