Estimate the area between the curve y = xand the lines y = 1 and y = 2.

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Solution:

Given

Given Curves are

y = x2         — (i)

y = 1          — (ii)

y = 2          — (iii)

Clearly (i) is a parabola that has a vertices at (0, 0) and the line of symmetry is x = 0 with y  0.

Also, (ii) and (iii) are the straight lines that are parallel to x-axis. With these information, the sketch in figure, in which the bounded region by (i), (ii), (iii) is the shaded portion.

User Loaded Image | CSIT Guide

Clearly the region has two symmetrical parts I and II. From figure, part – I, y moves from 1 to 2.

Now, the area of the bounded region is

\(\)

\(Area = 2\left | \int_{y = 1}^{2} x dy \right |\)

= \(2\left | \int_{1}^{2} \sqrt{y} dx \right | \)

= \(2\left [ \frac{y^{\frac{3}{2}}}{\frac{3}{2}} \right ]^2_1\)

= \(\frac{4}{3} (2^{\frac{3}{2}} – 1^{\frac{3}{2}})\)

= 3.77 sq. units

Thus, the area of the region is 3.77 sq. units

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