What is the area of the region between the graphs of f(x)=(4)/(x),g(x)=4, and x=4 ? Choose 1 answer: (A) 24-4ln(3) (B) 24-4ln(4) (C) 12-4ln(3) (D) 12-4ln(4)

Question

What is the area of the region between the graphs of  f(x)=(4)/(x),g(x)=4, and  x=4 ? Choose 1 answer: (A)  24-4ln(3) (B)  24-4ln(4) (C)  12-4ln(3) (D)  12-4ln(4)

Bytelearn · Accepted Answer

Intersection Point Calculation: To find the area between the curves, we need to set up an integral from the left boundary of the region to the right boundary. The left boundary is where the two functions intersect, and the right boundary is given as x=4. First, we need to find the intersection point of f(x) and g(x).Setting up Integral: The intersection point occurs where f(x) = g(x). So we set (4/x) = 4 and solve for x. 4/x = 4 4 = 4x x = 1 The functions intersect at x=1.Integration: Now we can set up the integral from x=1 to x=4. The area A between the curves is the integral of the top function minus the bottom function from x=1 to x=4. A = ∫ from x=1 to x=4 of (g(x) - f(x)) dx A = ∫ from x=1 to x=4 of (4 - 4/x) dxEvaluate Integral: We can now integrate the function from x=1 to x=4. A = ∫ from x=1 to x=4 of 4 dx - ∫ from x=1 to x=4 of 4/x dx A = [4x] from x=1 to x=4 - [4ln|x|] from x=1 to x=4Final Area Calculation: Evaluating the integrals, we get: A = (4*4 - 4*1) - (4ln|4| - 4ln|1|) A = (16 - 4) - (4ln(4) - 4ln(1)) Since ln(1) = 0, we simplify further. A = 12 - 4ln(4)We have found the area of the region between the graphs of f(x)=(4)/(x), g(x)=4, and x=4. The final answer is A = 12 - 4ln(4), which corresponds to answer choice (D).

What is the area of the region between the graphs of $f(x)=\frac{4}{x}, g(x)=4$ , and $x=4$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $24-4 \ln (3)$ $\newline$ (B) $24-4 \ln (4)$ $\newline$ (C) $12-4 \ln (3)$ $\newline$ (D) $12-4 \ln (4)$

Full solution

More problems from Write a quadratic function from its x-intercepts and another point

Related topics