# Find the area under the curve y = 2x^2 - 4x + 1 in [1, 3].

## Question:

Find the area under the curve {eq}y = 2x^2 - 4x + 1 {/eq} in {eq}[1, 3] {/eq}.

## Area Between the Curves:

The area under a curve between two points can be found by doing a definite integral between the two points.To find the area under the curve {eq}y = f(x) {/eq} between {eq}x = a {/eq} and {eq}x = b, {/eq} integrate {eq}y = f(x) {/eq} between the limits of {eq}a {/eq} and {eq}b {/eq} we use following formula

{eq}\displaystyle\mathrm{Area}=\int_{a}^{b} f(x) \,dx {/eq}

Consider the curves

{eq}\displaystyle y = 2x^2 - 4x + 1,\quad \left [ 1,3 \right ] {/eq}

The area between the curve is as follows

{eq}\displaystyle \mathrm{Area}=\int_{a}^{b} f(x) dx\\ \displaystyle =\int_{1}^{3} (2x^2 - 4x + 1)dx\\ \displaystyle =\left ( \frac{2}{3}x^3 - 2x^{2} + x \right )_{1}^{3}\\ \displaystyle =\frac{52}{3}-16+2\\ \displaystyle =\frac{10}{3} {/eq}