Find the area under the graph of f ( x ) = 3 2 x f\left(x\right)=3^{2x} between x = 0 x=0 and x = 2 x=2 .

this problem, we're asked to find the area under the graph of the function f of x equals three to the power of two x between x equals zero and x equals two. Now remember that the area under a curve is given by the integral. So here we wanna take the definite integral from zero to two, those bounds that we were given, of our function three to the power of two x d x. So let's go ahead and get started here. Now I know that I need to do a substitution because I have a function of x in that exponent. So doing my substitution here, I wanna set u equal to two x. This makes du equal to two times d x. Now, right now I can see that we're missing that constant of two. So that means that I need to multiply by that constant and its reciprocal to make this substitution work. Now, even though this is a definite integral, one of the methods that we can use here is to simply ignore those bounds until the very end after we've completed our substitution and that's the method I'm going to use here. So this is the integral of three to the power of two x, then I'm going to multiply by that two. So I have two d x, and if I multiply by two, I also need to multiply by one half. I'm going to go ahead and stick that on the outside of my integral here. And now looking at this, I have u and I have du. So this is equal to one half times the integral of three to the power of u du. Now having done substitutions with these exponential functions before I know that my rule here, this is going to give me one half times three to the power of u over the natural log of three. Then, of course, plus our constant of integration c, but we are actually working with a definite integral here. So instead, I can go ahead and replace that U with my original function of X. So here, two X, and bound it from zero to two. Now I can just plug in my bounds. So I'm gonna keep my constants out front. So keeping that one times two, a natural log of three out front, all of those are just constants. And then I'm just gonna focus on plugging my bounds in to this three to the power of two x. So this is three to the power of two times two minus three to the power of two times zero. Now two times zero is zero. So that makes this three to the power of zero, which is just one. Then three to the power of two times two. That is three to the power of four. So I have three to the power I have one over two times the natural log of three times three to the power of four minus one. Now what is three to the power of four? It is 81. So this is really 81 minus one over two natural log of three. 80 one minus one is of course 80 so this is really 80 over two natural log of three. Now we can do some further simplification here by dividing 80 by two to give us 40. So this is 40 over the natural log of three. This is an appropriate final answer here if you are not using a calculator in your course, but if you are using a calculator and you're expected to put this in decimal form, when you type this into your calculator you will get a final answer here of 36.41. Now let us know if you have any questions in the comments, and let's keep going.

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0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
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10. Physics Applications of Integrals 3h 16m
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16. Parametric Equations & Polar Coordinates7h 58m

11. Integrals of Inverse, Exponential, & Logarithmic Functions

Integrals of Exponential Functions

Calculus

11. Integrals of Inverse, Exponential, & Logarithmic Functions

Integrals of Exponential Functions

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Multiple Choice

Find the area under the graph of $f (x) = 3^{2 x}$ between $x equals 0$ and $x equals 2$ .

36.41

12.29

36.86

12.98

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Verified step by step guidance

Step 1: Recognize that the problem is asking for the area under the curve of the function f(x) = 3^(2x) between x = 0 and x = 2. This requires calculating the definite integral of f(x) from 0 to 2.

Step 2: Write the integral expression: $ \int_{0}^{2} 3^{2x} \, dx $. The goal is to evaluate this integral.

Step 3: Recall that the integral of an exponential function of the form $ a^{kx} $ is $ \frac{a^{kx}}{k \ln(a)} $, where $ a > 0 $ and $ a \neq 1 $. Here, $ a = 3 $ and $ k = 2 $.

Step 4: Apply the formula to integrate $ 3^{2x} $: $ \int 3^{2x} \, dx = \frac{3^{2x}}{2 \ln(3)} + C $, where $ C $ is the constant of integration. For a definite integral, the constant $ C $ is not needed.

Step 5: Evaluate the definite integral by substituting the limits of integration (x = 2 and x = 0) into the antiderivative: $ \left[ \frac{3^{2x}}{2 \ln(3)} \right]_{0}^{2} = \frac{3^{4}}{2 \ln(3)} - \frac{3^{0}}{2 \ln(3)} $. Simplify this expression to find the area under the curve.