Home > Calculus > Calculating Areas using Integrals

How do you find the area under the curve #f(x)=x^2# on the interval [-1,1]?

Answer 1

Samantha Adkison

#A = 2/3# units squared

Area under a curve on an interval is analogous to the integral of the function of the curve on an interval.

#A = \int_(x_i)^(x_f)f(x)dx#

#= \int_(-1)^(1) x^2 dx#

#=[x^3/3]_(x=-1)^(x=1)#

#=1/3 - (-1/3) = 2/3# units squared

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

Answer 2

Charles Arocha

To find the area under the curve (f(x) = x^2) on the interval ([-1, 1]), you can integrate the function over that interval using the definite integral formula. So, integrate (x^2) from -1 to 1 with respect to (x). The integral of (x^2) with respect to (x) is (\frac{x^3}{3}). Evaluate this expression from -1 to 1, then subtract the value of the integral at the lower limit from the value at the upper limit to find the area under the curve.

[ \int_{-1}^{1} x^2 dx = \left[\frac{x^3}{3}\right]_{-1}^{1} = \left(\frac{1^3}{3}\right) - \left(\frac{(-1)^3}{3}\right) = \frac{1}{3} - \left(-\frac{1}{3}\right) = \frac{2}{3} ]

So, the area under the curve (f(x) = x^2) on the interval ([-1, 1]) is (\frac{2}{3}).

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.