How do you evaluate the integral #int_0^1x^2dx# ?

Answer 1
#int_0^1 x^2 dx = 1/3#

Remember one of the most important theorems in Calculus:

Fundamental Theorem of Calculus (Part 2)

#int_a^b f(x)dx=[F(x)]_a^b=F(b)-F(a)#,
where #F(x)# is an antiderivative of #f(x)#.

Let us the theorem above to evaluate the definite integral.

#int_0^1 x^2 dx#
by finding an antiderivative of #x^2# using Power Rule,
#=[x^3/3]_0^1#

by plugging in the upper and the lower limits,

#=(1)^3/3-(0)^3/3#

by simplifying,

#=1/3#
Sign up to view the whole answer

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

Sign up with email
Answer 2

To evaluate the integral (\int_0^1 x^2 , dx), you can use the fundamental theorem of calculus, which states that the integral of a function over an interval can be found by evaluating its antiderivative at the endpoints of the interval and taking the difference.

  1. Find the antiderivative of (x^2).
  2. Evaluate the antiderivative at the upper limit of integration (1).
  3. Evaluate the antiderivative at the lower limit of integration (0).
  4. Subtract the value at the lower limit from the value at the upper limit.

[ \int_0^1 x^2 , dx = \left[ \frac{1}{3} x^3 \right]_0^1 = \frac{1}{3}(1^3) - \frac{1}{3}(0^3) = \frac{1}{3} ]

Sign up to view the whole answer

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

Sign up with email
Answer 3

To evaluate the integral ∫₀¹x² dx, we use the power rule for integration. Integrating x² with respect to x yields (1/3)x³. Evaluating this expression from 0 to 1 gives [(1/3)(1)³] - [(1/3)(0)³]. Simplifying, we get (1/3) - 0, which equals 1/3. Therefore, the value of the integral ∫₀¹x² dx is 1/3.

Sign up to view the whole answer

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

Sign up with email
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.

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.

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.

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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7