# Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or...

## Question:

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region. {eq}2y=5x^{(1/2)} {/eq} , y=4 and 2y+x=6

## Definite integral as area of a region between two curves:

To calculate the area of a flat region defined by curves, definite integrals and the fundamental theorem of the calculation are used. The procedure is illustrated below.

If {eq}\displaystyle g(y) {/eq} and {eq}\displaystyle h(y) {/eq} are continuous, not negative and {eq}\displaystyle g(y)\geq h(y) {/eq}

in the closed interval {eq}\displaystyle [c,d] {/eq}, then the area of the region bounded by the graph of {eq}\displaystyle g(y) {/eq}, {eq}\displaystyle h(y) {/eq} and the horizontal lines {eq}\displaystyle y=c {/eq} and {eq}\displaystyle y=d {/eq} is given by

{eq}\displaystyle A=\int_c^d[g(y)-h(y)]dy {/eq}

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Functions:

{eq}\displaystyle 2y=5x^{\frac{1}{2}}\\ \displaystyle \frac{2}{5}y=x^{\frac{1}{2}}\\ \displaystyle... The Fundamental Theorem of Calculus

from

Chapter 12 / Lesson 10
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The fundamental theorem of calculus links derivatives and antiderivatives in order to find the area under a curve. Learn more about the theorem with an example using velocity.