Basic Calculus Questions - polar integration and roots

[perihelion]

It's been a while since I studied calculus and basically I have a review sheet for a course I'm taking, but not a graded assignment. So, I was hoping if anyone knew a resource to point me in the right direction with a couple of problems:

<br /> \int_0^\theta x^a dx<br />

Where a is not an element of the set {0, -1} and theta > 0. I'm going over my old calculus text but still at a loss as to how to set this problem up, but maybe I'm making it more difficult than it is.

Also, I had a question involving finding real values of x such that a given function is 0, and justifying my answer. I just did this algebraically, but was wondering if there was another way that I've forgotten using calculus:

<br /> f(x) = x^2\ +\ 2*x\ +\ 2<br />
<br /> (x+1)*(x+1)+1 = 0<br />
<br /> (x+1)^2 = -1<br />
<br /> x = -1 - i, -1 + i<br />

Thanks.

 

[Mark44]

The antiderivative of x^a is 1/(a + 1) x^{a + 1}.

 

[perihelion]

The antiderivative of x^a is 1/(a + 1) x^{a + 1}.

Thanks, but I was feeling that maybe I was overlooking something. At this point I just stopped.

<br /> <br /> \int_0^\theta x^a dx\ =\ (\theta^(a+1))/(a\ +\ 1)<br /> <br />

 

[Mark44]

Thanks, but I was feeling that maybe I was overlooking something. At this point I just stopped.

<br /> <br /> \int_0^\theta x^a dx\ =\ (\theta^(a+1))/(a\ +\ 1)<br /> <br />

I think you meant this as \theta^{a + 1}/(a + 1)
but it didn't quite come out that way.

 

[Mark44]

For your other problem, find the zeroes of f(x) = x² + 2x + 2, the algebraic (as opposed to calculus) way is one way to go. In fact, calculus is not really appropriate for this type of problem.

f(x) = x² + 2x + 2 = x² + 2x + 1 + 1 = (x + 1)² + 1

The squared term is always >= 0, so adding 1 gives a value that is always >= 1, hence there are no real values for which f(x) = 0.

Looking at this graphically, the function's graph is a parabola that opens up, and whose vertex is at (-1, 1). Since the vertex is the lowest point on the graph, and it is above the x-axis, there are no values of x for which f(x) is less than or equal to zero.