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How do you find the area of the region under the curve #y=4x^-2# from x=1, to #x=oo#?

Answer 1

Samantha Adkison

Area= #4# sq. units

Area= #int_1^oo 4x^-2#

Area= #[-4x^-1]_1^oo#

Area= #0-(-4)#

Area= #4# sq. units

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Answer 2

Adam Adkins

To find the area under the curve (y = 4x^{-2}) from (x = 1) to (x = \infty), you need to evaluate the definite integral of the function from (x = 1) to (x = \infty). The integral is given by:

[A = \int_{1}^{\infty} 4x^{-2} , dx]

This integral represents the area under the curve between the limits of (x = 1) and (x = \infty). To solve this integral, you can use calculus techniques. In this case, since the function is of the form (4x^{-2}), you can integrate it using the power rule of integration:

[A = \lim_{b \to \infty} \int_{1}^{b} 4x^{-2} , dx]

[= \lim_{b \to \infty} \left[-4x^{-1}\right]_{1}^{b}]

[= \lim_{b \to \infty} \left(-4\left(\frac{1}{b}\right) + 4\right)]

[= 4]

Therefore, the area under the curve (y = 4x^{-2}) from (x = 1) to (x = \infty) is (4).

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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.