How to solve this using integration ?

Question

The curve #2y = x^3# and straight line #y = 2x# crossing at three dots, O, A and B.

(a) Find a limited area between curves #2y = x^3# with OA cores ( The answers : #2 unit^2# )

(b) Hence, find the area of the shaded region bounded by the curve #2y = x^3# and the straight line #y = 2x# ( The answers : #4 unit^2# )

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Answer 1 · 2018-02-26T13:49:17

See the explanation below

We need

#intx^ndx=x^(n+1)/(n+1)+C (n!=-1)#

#"PART (a)"#

The limited area is

#dA_1=(y_1-y_2)dx=(2x-x^3/2)dx#

#A_1=int_0^2(2x-x^3/2)dx#

#=[x^2-x^4/8]_0^2#

#=(4-16/8)-(0)#

#=4-2=2u^2#

#PART (b)#

Either proceed by symmetry or perform the calculation

#A_2=int_-2^0(2x-x^3/2)dx#

#=[x^2-x^4/8]_-2^0#

#=(0)-(4-16/8)#

#=-2u^2#

Hence,

#|a_2|=2u^2#

The total shaded area is #=2+2=4u^2#