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How do you evaluate the definite integral #int x^2 dx# from #[1,2]#?

Answer 1

Christopher Allers

#int_1^2x^2dx=7/3#

As #d/(dx)x^3=3x^2#

#int_1^2x^2dx=|x^3/3|_1^2#

= #(2^3/3-1^3/3)#

= #8/3-1/3=7/3#

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Answer 2

Isaiah Allen

To evaluate the definite integral (\int_{1}^{2} x^2 , dx), you need to find the antiderivative of (x^2), then evaluate it at the upper limit (2) and subtract the value at the lower limit (1).

The antiderivative of (x^2) is (\frac{1}{3}x^3).

Evaluate (\frac{1}{3}x^3) at (x = 2) and (x = 1):

(\frac{1}{3}(2)^3 = \frac{8}{3})

(\frac{1}{3}(1)^3 = \frac{1}{3})

Subtract the value at the lower limit from the value at the upper limit:

(\frac{8}{3} - \frac{1}{3} = \frac{7}{3})

So, (\int_{1}^{2} x^2 , dx = \frac{7}{3}).

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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.