Estimate the area between the curve y = x2 and the lines y = 1 and y = 2.

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Suresh Chand · Accepted Answer

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. 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 = 2left | int_{y = 1}^{2} x dy right |) = (2left | int_{1}^{2} sqrt{y} dx right | ) = (2left [ 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

Estimate the area between the curve y = x²and the lines y = 1 and y = 2.

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Estimate the area between the curve y = x2 and the lines y = 1 and y = 2.

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Estimate the area between the curve y = x²and the lines y = 1 and y = 2.