Find the area under the given curves and given lines:
(i)y=x2,x=1,x=2 and x-axis
(ii)y=x4,x=1,x=5 and x-axis
Find the area under the given curves and given lines:
(i)y=x2,x=1,x=2 and x-axis
(ii)y=x4,x=1,x=5 and x-axis
Solution and Explanation
i. The required area is represented by the shaded area ADCBA as
Area ADCBA=
\[\int_{1}^{2} y \,dx\]=\(\int_{1}^{2} x^2 \,dx\)
=[\(\frac{x^3}{3}\)]21
=\(\frac{8}{3}\)-\(\frac{1}{3}\)
=\(\frac{7}{3}\)units
ii. The required area is represented by the shaded area ADCBA as
Area ADCBA=∫01x4dx
=[\(\frac{x^5}{5}\)]51
=\(\frac{(5)^5}{5}\)-\(\frac{1}{5}\)
=(5)4-\(\frac{1}{5}\)
=625-\(\frac{1}{5}\)
=624.8units.
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