Mathguru – Help – Find the area between two given curves.

8.2 Area Between Two Curves > No. 2

Find the area between two given curves.

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Explanation:

Integral

A definite integral of a function can be represented as the signed area of the region bounded by its graph.

Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral

$Description: \int_a^b \! f(x)\,dx \,$

is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b.

The term integral may also refer to the notion of antiderivative, a function F whose derivative is the given function ƒ. In this case, it is called an indefinite integral and is written:

$Description: F = \int f(x)\,dx.$

Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if ƒ is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of ƒ is known, the definite integral of ƒ over that interval is given by

$Description: \int_a^b \! f(x)\,dx = F(b) - F(a)\,$

(Our solved example in mathguru.com uses this concept)

http://en.wikipedia.org/wiki/Integral

List of integrals of irrational functions

Integrals involving $Description: u = \sqrt{a^2-x^2}$

$Description: \int u \;dx = \frac{1}{2}\left(xu+a^2\arcsin\frac{x}{a}\right) \qquad\mbox{(}|x|\leq|a|\mbox{)}$

(Our solved example in mathguru.com uses this concept)

$Description: \int xu\;dx = -\frac{1}{3} u^3 \qquad\mbox{(}|x|\leq|a|\mbox{)}$

$Description: \int x^2u\;dx = -\frac{x}{4} u^3+\frac{a^2}{8}(xu+a^2\arcsin\frac{x}{a}) \qquad\mbox{(}|x|\leq|a|\mbox{)}$

$Description: \int\frac{u\;dx}{x} = u-a\ln\left|\frac{a+u}{x}\right| \qquad\mbox{(}|x|\leq|a|\mbox{)}$

$Description: \int\frac{dx}{u} = \arcsin\frac{x}{a} \qquad\mbox{(}|x|\leq|a|\mbox{)}$

$Description: \int\frac{x^2\;dx}{u} = \frac{1}{2}\left(-xu+a^2\arcsin\frac{x}{a}\right) \qquad\mbox{(}|x|\leq|a|\mbox{)}$

$Description: \int u\;dx = \frac{1}{2}\left(xu-\sgn x\,\operatorname{arcosh}\left|\frac{x}{a}\right|\right) \qquad\mbox{(for }|x|\ge|a|\mbox{)}$

$Description: \int \frac{x}{u}\;dx = -u \qquad\mbox{(}|x|\leq|a|\mbox{)}$

http://en.wikipedia.org/wiki/List_of_integrals_of_irrational_functions

The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.