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Application of Integrals

In this chapter, we will study applications of integrals to find areas bounded by curves and areas between conic sections like circles, parabola, ellipses, etc. Topics which include - areas under simple curves, especially lines, areas of conic sections in standard form only, the region should be clearly identifiable.

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Exercise 1

  • Q1

    Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.

    Ans:

    The area of the region bounded by the curve, y2 = x, the lines, x = 1 and x = 4, and the x-axis is the area ABCD.

    \begin{align}Area \;of\; ABCD = \int_{1}^{4} y.dx \end{align}

     \begin{align} = \int_{1}^{4} \sqrt{x}.dx \end{align}

     \begin{align} =\left[\frac{x^\frac{3}{2}}{\frac{3}{2}}\right]_1^4 \end{align}

     \begin{align} =\frac{2}{3}\left[(4)^\frac{3}{2} - (1)^{\frac{3}{2}}\right] \end{align}

    \begin{align} =\frac{2}{3}\left[8 -1\right] \end{align}

    \begin{align} =\frac{14}{3}\; Units \end{align}


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