Integrals and Sigma

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December 3rd 2012, 07:15 PM
sean2012

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Integrals and Sigma

Hello I need help with 2 problems.

The first is

Code:

F(x)= Integral from sqrt(x) to 2 (tan(t)/t^2) dt Find F'(x)

The second is

Code:

Find the actual area under the curve f(x)=2x^2 on the interval [0,3]. Use right hand endpoints and the formula Area = lim n→ ∞ n Σ i=1 f(x^*i) delta(x)

Here is an image of the equations
Attachment 26043
December 3rd 2012, 07:25 PM
Prove It

Re: Integrals and Sigma

For the first part, I would perform the following transformation:

$\displaystyle \begin{align*} \int_2^{\sqrt{x}}{\frac{\tan{(t)}}{t^2}\,dt} &= \frac{1}{2}\int_2^{\sqrt{x}}{\frac{2t\tan{(t)}}{t^ 3}\,dt} \end{align*}$

Let $\displaystyle \begin{align*} u = t^2 \implies du = 2t\,dt \end{align*}$ and changing the bounds gives

$\displaystyle \begin{align*} \frac{1}{2} \int_4^x { \frac{ \tan{ \left( \sqrt{u} \right) } }{ u^{\frac{3}{2}} }\,du } \end{align*}$

Then you can make use of the Second Fundamental Theorem of Calculus.

For the second, you have been given a formula to use. Where exactly are you stuck?
December 3rd 2012, 07:30 PM
sean2012

Re: Integrals and Sigma

Okay thanks ill take another look at the first problem and for the second problem the equation confuses me. What does the x*i mean?
December 3rd 2012, 07:33 PM
Prove It

Re: Integrals and Sigma

$\displaystyle \begin{align*} x^*_i \end{align*}$ is just a shorthand to give a label to each value of x.
December 3rd 2012, 08:01 PM
sean2012

1 Attachment(s)

Re: Integrals and Sigma

Okay I get problem 2 all the way up to this point

Attachment 26047

What is the n^3/3 + ...