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Chapter 12: Problem 1048

If \(a<0

Short Answer

Expert verified
The short answer is: \(b\int_a^b\frac{|x|}{x} dx = ab + b^2\)

Step by step solution

01

Rewrite the function with the absolute value function

The given function is: \(|x|/x\) We can rewrite the function without using the absolute value by considering the cases when x is positive or negative. If x < 0, then |x| = -x, and |x|/x = -1 If x > 0, then |x| = x, and |x|/x = 1 Now, we rewrite the function as: \[f(x) = \begin{cases} -1 & \text{if } x < 0 \\ 1 & \text{if } x \geq 0 \end{cases} \]
02

Break the integral using the cases

Now we will break the integral into two parts using the function definition above and the given bounds (a, b). Since a < 0 and b > 0, the integral can be split at x = 0: \[b\int_a^b\frac{|x|}{x} dx = b\int_a^0\frac{-x}{x} dx + b\int_0^b\frac{x}{x} dx\]
03

Evaluate the integrals

Now, we evaluate the integrals for each part: \[b\int_a^0\frac{-x}{x} dx = b\int_a^0(-1) dx = -b[x]_a^0 = -b(0-a) = ab\] \[b\int_0^b\frac{x}{x} dx = b\int_0^b 1 dx = b[x]_0^b = b(b-0) = b^2\]
04

Combine the results and simplify

Now, we combine the results from step 3 and simplify: \(b\int_a^b\frac{|x|}{x} dx = ab + b^2\)
05

Identify the correct option

Now, we check the given multiple-choice options to find the correct answer: (a) a + b: This cannot be the correct answer since our result contains an extra b term. (b) b - a: This also cannot be the correct answer since our result contains a squared term. (c) a - b: This cannot be the correct answer since our result contains a squared term. (d) \((b - a) / 2\): This also cannot be the correct answer since our result does not divide the expression by 2. None of the options match the result we obtained (ab + b^2). There is a possibility of an error in the given options, so we conclude with our obtained result: \[b\int_a^b\frac{|x|}{x} dx = ab + b^2\]

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Most popular questions from this chapter

The value of integral \({ }^{1} \int_{0}\left[1 /\left\\{1-\mathrm{x}+\sqrt{\left. \left.\left(2 \mathrm{x}-\mathrm{x}^{2}\right)\right\\}\right] \mathrm{d} \mathrm{x} \text { is...... }}\right.\right.\) (a) 1 (b) \((1 / 2)\) (c) \((\pi / 4)\) (d) \((\pi / 2)\)

The area enclosed by \(\mathrm{y}^{2}=32 \mathrm{x}\) and \(\mathrm{y}=\mathrm{m} \mathrm{x}(\mathrm{m}>0)\) is \((8 / 3)\) then \(\mathrm{m}\) is....... (a) 1 (b) 2 (c) 4 (d) \((1 / 4)\)

The area of common region of the circle \(x^{2}+y^{2}=1\) and \((x-1)^{2}+y^{2}=1\) is given by (a) \((2 \pi / 3)+\sqrt{(3 / 2)}\) (b) \((\pi / 3)-\sqrt{(3 / 2)}\) (c) \((\pi / 3)+\sqrt{(3 / 2)}\) (d) \((2 \pi / 3)-(\sqrt{3} / 2)\)

The value of the integral \(\left.{ }^{1} \int_{-1} \log \left[1 /\left\\{x+\sqrt{(} x^{2}+1\right)\right\\}\right] d x\) is \(\ldots \ldots\) (a) \(\log 2\) (b) 0 (c) \(\log 3\) (d) not possible

The area of the region bounded by curves \(x^{2}+y^{2}=4, x=1\) \(\& x=\sqrt{3}\) is..... (a) \((\pi / 3)\) sq. unit (b) \((2 \pi / 3)\) sq. unit (c) \((5 \pi / 6)\) sq. unit (d) \((4 \pi / 3)\) sq. unit
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