Evaluate the integral: ( int x^3 sqrt16-x^2 d x ) (A) Which...

Evaluate the integral: ( int x^{3} sqrt{16-x^{2}} d x ) (A) Which trig substitution is correct for this integral? ( x=4 sin ( heta) ) ( x=16 sin ( heta) ) ( x=16 sec ( heta) ) ( x=4 an ( heta) ) ( x=4 sec ( heta) ) ( x=16 an ( heta) ) (B) Which integral do you obtain after substituting for ( x ) and simplifying? Note: to enter ( heta ), type the word theta. ( int square d heta ) (C) What is the value of the above integral in terms of ( heta ) ? ( square ) ( +mathrm{C} ) (D) What is the value of the original integral in terms of ( x ) ? ( square ) ( +mathrm{C} )

Transcript

00:01 Here we have integral x cubed and then we have a square root 16 minus x squared.

00:06 So what i'm going to do here is i'm going to say that x is 4 sine theta so that dx is 4 cosine d theta.

00:22 And so what that looks like is if we have this triangle here, this is going to be radical 16 minus x squared.

00:37 This here is going to be x.

00:40 This is going to be 4.

00:41 So see that sine theta equals x over 4.

00:47 So this is going to be your answer to the first part.

00:52 And then when we plug that in we get this integral so it's going to be instead of x we're going to plug in 4 sine theta cubed then here we have radical so 16 and then instead of x squared.

01:14 We're going to write 4 sine theta squared and then dx is going to be 4 cosine, sorry this is theta, d theta.

01:29 So let's simplify this a little.

01:31 This is going to be 64 sine sine cubed theta square root.

01:38 Here we have 16 minus 16 sine squared theta.

01:43 I'm actually going to simplify this now.

01:46 This is basically the same as saying 16 parentheses 1 minus sine squared theta.

01:55 I will come back to that, but if you remember your trig identities, you'll know what to do there.

01:59 And this is 4 cosine theta d theta.

02:02 So if you remember your trig identities, sine squared theta plus cosine squared theta equals 1.

02:10 So this is equal to cosine theta.

02:14 Let's write that down.

02:15 I'm going to bring the 4, multiply it to the 64, and then i'm going to bring out another 4 from here.

02:22 So this is going to be 64 times 4 times 4 sine cubed theta square root of, this is cosine squared theta.

02:36 This is cosine theta d theta...

Question

Please give Ace some feedback