Question:
Find the area of the region bounded by curves \(y=x^2+2,y=x,x=0\) and \(x=3\)
Find the area of the region bounded by curves \(y=x^2+2,y=x,x=0\) and \(x=3\)
Updated On: Sep 18, 2023
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Solution and Explanation
The correct answer is:\(\frac{21}{2}units.\)
The area bounded by the curves,\(y=x^2+2,y=x,x=0,\)and \(x=3\),is represented by
the shaded area OCBAO as
Then,Area OCBAO=Area ODBAO-Area ODCO
\(=∫^3_0(x^2+2)dx-∫^3_0xdx\)
\(=[\frac{x^3}{3}+2x]^3_0-[\frac{x^2}{2}]^3_0\)
\(=[9+6]-[\frac{9}{2}]\)
\(=15-\frac{9}{2}\)
\(=\frac{21}{2}units.\)
The area bounded by the curves,\(y=x^2+2,y=x,x=0,\)and \(x=3\),is represented by
the shaded area OCBAO as
Then,Area OCBAO=Area ODBAO-Area ODCO
\(=∫^3_0(x^2+2)dx-∫^3_0xdx\)
\(=[\frac{x^3}{3}+2x]^3_0-[\frac{x^2}{2}]^3_0\)
\(=[9+6]-[\frac{9}{2}]\)
\(=15-\frac{9}{2}\)
\(=\frac{21}{2}units.\)
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Concepts Used:
Area between Two Curves
Integral calculus is the method that can be used to calculate the area between two curves that fall in between two intersecting curves. Similarly, we can use integration to find the area under two curves where we know the equation of two curves and their intersection points. In the given image, we have two functions f(x) and g(x) where we need to find the area between these two curves given in the shaded portion.
Area Between Two Curves With Respect to Y is
If f(y) and g(y) are continuous on [c, d] and g(y) < f(y) for all y in [c, d], then,
