Math  /  Calculus

QuestionSet up the definite integral that gives the area of the shaded region. Do not evaluate the integral
The definite integral is 6(dx\int_{\square}^{6}(\square \mathrm{dx}.

Studdy Solution

STEP 1

What is this asking? We need to find the definite integral that represents the area under the curve y=2xy = \frac{2}{x} between x=3x = 3 and x=6x = 6. Watch out! Make sure you put the lower and upper limits of integration in the correct places!
Also, remember we're *not* evaluating the integral, just setting it up.

STEP 2

1. Identify the function.
2. Determine the limits of integration.
3. Set up the definite integral.

STEP 3

The function that defines the curve is already given to us: y=2xy = \frac{2}{x}.
This tells us how the height of the shaded region changes as xx changes.
Awesome!

STEP 4

The shaded region starts at x=3x = 3 and ends at x=6x = 6.
These are our **limits of integration**.
The **lower limit** is x=3x = \textbf{3}, and the **upper limit** is x=6x = \textbf{6}.
These tell us the starting and ending points of the area we're interested in.

STEP 5

The definite integral representing the area under a curve f(x)f(x) from x=ax = a to x=bx = b is given by abf(x)dx\int_{a}^{b} f(x) \, \mathrm{dx}.
Here, our function is f(x)=2xf(x) = \frac{2}{x}, our **lower limit** aa is **3**, and our **upper limit** bb is **6**.

STEP 6

So, plugging in our values, we get the definite integral: 362xdx\int_{\textbf{3}}^{\textbf{6}} \frac{2}{x} \, \mathrm{dx}.
This integral represents the area of the shaded region!

STEP 7

The definite integral that gives the area of the shaded region is 362xdx\int_{\textbf{3}}^{\textbf{6}} \frac{2}{x} \, \mathrm{dx}.

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