Home > Calculus > Integration: the Area Problem

Let R be the region in the first quadrant bounded by the graphs of #y=x^2#, #y=0#, and #x=2#, how do you find the area of R?

Answer 1

Ezekiel Alexander

#R = 8/3#

#R = int_0^2 x^2dx = x^3/3|_0^2 = 8/3#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Elijah Abram

To find the area of region ( R ), bounded by the graphs of ( y = x^2 ), ( y = 0 ), and ( x = 2 ) in the first quadrant, you can use the definite integral. The area ( A ) of ( R ) is given by:

[ A = \int_{0}^{2} (x^2 - 0) , dx ]

Evaluating this definite integral yields:

[ A = \int_{0}^{2} x^2 , dx = \left[ \frac{x^3}{3} \right]_{0}^{2} = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3} ]

So, the area of region ( R ) is ( \frac{8}{3} ) square units.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.