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Use Theorem 5.5 .5 to evaluate the given integrals.$$\begin{array}{llll}{\text { (a) } \int_{-1}^{1}|2 x-1| d x} & {\text { (b) } \int_{0}^{3 \pi / 4}|\cos x| d x} & {} & {}\end{array}$$

(a) $\frac{5}{2}$(b) $2-\frac{\sqrt{2}}{2}$

Calculus 1 / AB

Calculus 2 / BC

Chapter 5

INTEGRATION

Section 6

The Fundamental Theorem of Calculus

Functions

Limits

Differentiation

Integrals

Integration

Integration Techniques

Continuous Functions

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always the hardest problem. There are two parts and B. They are dealing with absolute value functions. So we're gonna have to split the into gloom, right? Someone function crosses the X axis, uh, or absolute value functions they for give you example of the actual value of X Instead of crossing over the function like that, it would be like this. So we're actually need to integrate positive values or need sign. All right, so, uh, who were looking our phone first part? It would be two x minus one. So we need to find when two X minus one equals zero and equals zero at X equals to one half, because when you plug and one happened to two experts when you get zero, and this is indeed between the interval in eight of 1 to 1. So we need to split this function up into two parts. The first would be the integral from native 1 to 1 half of two x minus one DX and second one would be to go from one half to one of two X minus one. The X and you are no longer absolute values, just parentheses. People. Quick fix them Now we have to consider which ones we have to make negative. So the ones that we're gonna make negative is going to be the one the stereo, The interval wearing the function is below the X axis. Since that would be normal. However, the absolute value changes for the function to be above changes to swap, to flip over to the positive. Why access? So which one would be normally below the X axis? Well, if you plugged in, made of one into our function to recognize when you get negative two minus one, which is 93. So that is indeed below the X axis. So this would be negative and the other would be positive since it is above the X axis. So evaluating these into Earl's uh, um, this all the way evaluating these intervals we get would get negative of your head rivet of two x minus One is going to be X squared minus X on is from made of 11 half founded WAAS Uh, you intended route of up to recognize one's again X squared minus X founded by one half to one. So would a plug in our variables. Her ex we get, We get negative times 1/4 minus one half minus. They want to be one minus. Name one plus one. So I will be our first one. Waas, uh, actually minus x 01 minus one or one minus one house. Where the 1/4 minus one half. All right, so if we were to So all this out we get, we'd end up, we'd end up with an answer off five house. So that's for part A Now, for part B, we're dealing with off coast science function, right? So when does the coast and function equal to zero? Uh, well, within the range from 0 to 3. Pirate for those kind of X equals 01 of her ex equals pi over two. Right. We want experts part material. Whole sign of the function equals 20 So now we're gonna have to split our function from zero to part of it too. Both sigh, Novaks, dx and then the other, probably from pirate, too. Three pie forms. Of course I d x, of course, that xcx so which one would be negative is gonna be the part of the function that is below the X axis. And we know that co sign starts off above the X axis and then becomes a negative function after pi over two. So it would be a positive for first into go and then negative for the second into girl. All right, so if we were to evaluate, if you were to evaluate this thing to go, we get the integral of coastline of exit. Just the science functions would have sign of acts from zero Zafira too minus the integral clothes. And against will be signed off X from part of a tomb 23 pie Ford's. So if you were plugging our variables for X, we get sign up, I over to sign of pi over two minus sign of zero minus sign of three pi over fourths. Um, since we nine minus is that we have made of simple right, very become positive all the plus sign of pi over two. All right, so we know that sign up, however, to is going to be one minus sign of zero is zero. In a sign of three Piper, four seats out of three, power for your baby herded to over to and finally sounded pirate too is going to be one, So our final answer is going to be two minus. It's weird to over to.

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