Find the area under the curve, of the function f ( x ) = 3 x + 2 f(x)=3x+2 from x = 0 x=0 to x = 4 x=4 using limits.

see how we can solve this problem. So we're asked to find the area under the curve of this function right here from x equals 0 to x equals 4 using limits. So this is a situation where we're finding the exact area, so we're no longer approximating. Now to do this, there are a few equations that we need. First up, if you want to calculate the area, the area is going to be equal to the limit as n approaches infinity, and that's going to be the summation k equals 1 to n for delta x times f of x k. Now we also need this equation when we're calculating the xk here. Xk is going to be equal to a plus k times delta x. And then another equation that we'll need is this one, delta x is equal to b minus a over n. So these are the equations we can use to solve this problem, where this is really the main equation that we're trying to plug everything into. So let's go ahead and do this. Now always I I feel that always a good way to start any of these problems is to start by calculating the base. Now the base, we can use these values and the number of rectangles, which is going to be infinite since we're using limits. So delta x, which we have over here, is going to be equal to b, the end of our interval, which is 4, minus a, the start of our interval, which is 0, all divided by n. Now because n is becoming infinitely large, we're using limits, meaning that n is going to infinity, we're just going to keep n as a variable since we can't really plug infinity into an equation. So we're going to have 4 over n, and that is the base. So using this equation up here, what I can do is write this as the area is equal to the limit as n approaches infinity, and then we're going to have the summation k equals 1 to n. And then we figured out delta x is 4 over n. And now what we need to do is figure out what f of k of x is, or f of x k, I should say. Now to find f of x k, recall that x k is a plus k times delta x. So we can use the equations that we have. We can see that a is 0, and this is going to be plus k times the delta x we just calculated, 4 over n. Now if we go ahead and simplify this, it's going to be 4 k over n because this 0 here, we don't need to write it. So this is going to be our x of k. So going back up to this this area that we have right here, this equation, we can replace f of x k, we can replace this x k in here with 4 k over n. So we have 4k over n in for this function. Now what I'm going to do is I'm going to take this 4k over n and plug it into our equation. And keep in mind, all of this information right here, you're going to want to keep writing for each step that we have. But to save some time, I'm just gonna put this in quotes here and say this continues on as we're calculating the area. So the area is going to equal this entire argument right here, and we're going to have 4 over n. And then plugging 4 k over n in for this equation right here, well, that's going to give us 3 times 4 k over n, and then that's going to be plus 2. So to simplify this, we're going to have 4 over n that's going to stay right there, and then we're going to have 3 times 4, which is 12. So we're gonna have 12 k over n, and then that's going to be plus 2. Now from here, what I can do is actually distribute this 4 over n into the parenthesis. So we're going to have 4 times 12, which will be 48 k over, and then n times n is n squared. And this is going to be plus 4 times 2, which is 8 over n. And then this right here is going to be all that algebra simplified. Keep in mind, this entire argument is still in front. So we're going to have that a is equal to the limit as n approaches infinity, and we're gonna have the summation k equals 1 to n, and we're going to have 48k over n squared. And then what I'm going to do is apply this summation twice. So we're gonna have the summation k equals 1 to n, and then we're going to have this next argument 8 over n. So this is what we have. Now at this point, since it's k that we're looking at, what I can do is take everything out of the summation that is not a k variable. So we'll keep this limit as n approaches infinity. What I'm going to have is 48 over n squared on the outside, and then we're going to have the summation, k equals 1 to n. And then this k variable that we have right there, plus, and then we're going to have 8 over n on the outside. We'll keep this summation k equals 1 to n, and then this will just be 1 since that's just a constant. Now from here, what I can do is apply the summation rules, and recall that when you see the summation of k, it's going to be n times n plus 1 over 2. So we'll keep this limit as n approaches infinity, and then we're going to have n times n plus 1 over 2. And then this is all gonna be multiplied by this, what we have on the outside here, so 48 over n squared. Then we'll have plus and any constant we have by itself, we just multiply it by n. So ultimately, this is just gonna become 8 n over n, where these n's are just going to cancel. So this is what we end up having. And so we're gonna keep this limit as n approaches infinity, and we're going to have n times n plus 1. I can cancel this n with one of those n's, meaning we're going to have 48 times n plus 1, and then that's going to be over 2 n, and then that's going to be this plus 8. Now from here, what I can do is apply this limit. And recall that for limits, as n approaches infinity, we need there are some certain rules that we can follow. Now notice the highest degree n we have is n to the first power on top, and we have an n to the first power on bottom. So because of this, we need to find a ratio of the coefficients out in front. If I were to distribute this 48, and 48 will be the coefficient, and then we'd have 2. So what we end up getting is that this whole area is equal to 48 over 2, which is the same thing as 24 +8. And 24 +8 is the same thing as 32. So your area is going to be equal to 32, and that right there is the solution to this problem. So that's how you can find the area underneath the curve using limits. Hope you found this video helpful, and let's move on.

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23. Intro to Derivatives & Area Under the Curve

Area Under a Curve

Precalculus

23. Intro to Derivatives & Area Under the Curve

Area Under a Curve

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Multiple Choice

Find the area under the curve, of the function $f (x) = 3 x + 2$ from $x equals 0$ to $x equals 4$ using limits.

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Verified step by step guidance

Step 1: Understand the problem. We need to find the area under the curve of the function f(x) = 3x + 2 from x = 0 to x = 4 using limits. This involves calculating the definite integral of the function over the given interval.

Step 2: Set up the integral. The area under the curve from x = 0 to x = 4 is given by the definite integral: $ \int_{0}^{4} (3x + 2) \, dx $.

Step 3: Find the antiderivative. The antiderivative of f(x) = 3x + 2 is F(x) = $ \frac{3}{2}x^2 + 2x $. This is the function whose derivative is f(x).

Step 4: Evaluate the definite integral using the Fundamental Theorem of Calculus. Substitute the upper and lower limits into the antiderivative: $ F(4) - F(0) $.

Step 5: Calculate the values. Compute F(4) and F(0) using the antiderivative found in Step 3, and subtract the two results to find the area under the curve.