fundamental theorem of calculus definite intergral problems

helppalusmath · #1 $Old$ December 1st, 2008, 07:19 PM

1) Find the definite integral from -10 to 10 if the equation is (100-x^2)dx?

2) f(x) = x if x<1
1/x if x>or equal to 1
Find the definite integral from -1 to 7 f(x)dx?

EightballLock · #2 $Old$ December 1st, 2008, 08:02 PM

Quote:

Originally Posted by helppalusmath $View Post$

1) Find the definite integral from -10 to 10 if the equation is (100-x^2)dx?

$\int_{-10}^{10}(100 - x^2)dx$
It'll be easier to think about if you break it into two integrals.
$\int_{-10}^{10}100dx - \int_{-10}^{10}x^2dx$
The first piece is easy enough to solve, and the second is power rule:
$F(x) = [100x - \frac{x^3}{3}]_{-10}^{10}$
Just solve $F(10) - F(-10)$ and you're set.

EightballLock · #3 $Old$ December 1st, 2008, 08:09 PM

Quote:

Originally Posted by helppalusmath $View Post$

2) f(x) = x if x<1
1/x if x>or equal to 1
Find the definite integral from -1 to 7 f(x)dx?

Piecewise defined functions are integrated in the same manner you would integrate separate functions.
$F(x) = \frac{x^2}{2} + C$ if $x < 1$
$F(x) = \ln{x} + C$ if $x \geq 1$
Just treat it as the sum of two integrals, the first piece from $(-1, 1)$ and the second piece from $(1, 7)$