Integrals and Sigma

• Dec 3rd 2012, 07:15 PM
sean2012
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
• Dec 3rd 2012, 07:25 PM
Prove It
Re: Integrals and Sigma
For the first part, I would perform the following transformation:

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

\displaystyle \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?
• Dec 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?
• Dec 3rd 2012, 07:33 PM
Prove It
Re: Integrals and Sigma
\displaystyle \displaystyle \begin{align*} x^*_i \end{align*} is just a shorthand to give a label to each value of x.
• Dec 3rd 2012, 08:01 PM
sean2012
Re: Integrals and Sigma
Okay I get problem 2 all the way up to this point

Attachment 26047

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