The most important application of Integrals involves finding areas bounded by a curve and x-axis. It includes findings solutions to the problems of work and energy.
In the next section, we present some examples of finding areas under the curve with the help of definite integrals.
`( 27 )/4`
We first find the points where the curve cuts the x-axis. We put `y = 0`
Therefore, the curve crosses the x-axis at (-3 , 0) and (0 , 0) … So we apply the integration limits from -3 to 0.
The required area
=`( 0 )/4 + 0–[ (–3)^(4 )/4+(–3)3 ]`` ( 2 )/3. [8 – 1] = ( 14 )/3`
The part of the curve above the x-axis is
`y = √(4-x)`
We put
` 4 – x = t `
` – dx = dt ` => ` dx = – dt `
So
For the value of `x = 0 => 4 – 0 = t => t = 4 `
And for `x = 3 => 4 – 3 = t => t = 1`
=>The Required Area Under the curve is given by
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