Shade the region bounded by f(x)=log2x+4\textcolor{blue}{f\left(x\right)=log_2x+4} & g(x)=ex−4\textcolor{orange}{g\left(x\right)=e^{x-4}} on the interval [1,4]. Then set up an integral to represent the region's area.

A = ∫ 1 4 ( l o g 2 x + 4 − e x − 4 ) d x A=\int_1^4\left(log_2x+4-e^{x-4}\right)dx A = ∫ 1 4 ( l o g 2 x + 4 − e x − 4 ) d x

Shade the region bounded by f ( x ) = l o g 2 x + 4 \textcolor{blue}{f\left(x\right)=log_2x+4} & g ( x ) = e x − 4 \textcolor{orange}{g\left(x\right)=e^{x-4}} on the interval [1,4]. Then set up an integral to represent the region's area.

this problem, we're asked to shade the region bounded by the two functions f of x equals log base two of x plus four and g of x equals e to the power of x minus four, specifically on the interval from one to four. We wanna set up an interval in order to represent this region's area. So let's first take a look at our graph view. We see our function f of x on the top in blue here and g of x in orange on the bottom. Now we're looking specifically at the area from one to four. So we don't wanna shade all of the area in between these two curves. We only wanna shade the area between one and four. So between one and four, we're just looking at that interior area there. So it's this area contained from one to four in between my two function curves. So with our area shaded, we now wanna set up an integral to represent this area. So remember that in order to find the area, we want a definite integral of our top function minus our bottom function. Now for this specific case, we know that our bounds should be from one to four because that's what's given to us in our problem. So So we have the definite integral from one to four of our top function. In this case, that's a log base two of x plus four minus our bottom function, which here is e to the power of x minus four, integrating with respect to x. So this is the integral that represents the area of this shaded region. If we actually wanted to find this area, we could go ahead and evaluate this integral, but because they just asked us to set it up here, we'll stop for now. If If you have any questions, feel free to let us know, and let's keep practicing.

Shade the region bounded by f ( x ) = l o g 2 x + 4 \textcolor{blue}{f\left(x\right)=log_2x+4} & g ( x ) = e x − 4 \textcolor{orange}{g\left(x\right)=e^{x-4}} on the interval [1,4]. Then set up an integral to represent the region's area.

A = ∫ 1 4 ( l o g 2 x + 4 − e x − 4 ) d x A=\int_1^4\left(log_2x+4-e^{x-4}\right)dx A = ∫ 1 4 ( l o g 2 x + 4 − e x − 4 ) d x

A = ∫ 1 4 ( l o g 2 x + 4 + e x − 4 ) d x A=\int_1^4\left(log_2x+4+e^{x-4}\right)dx A = ∫ 1 4 ( l o g 2 x + 4 + e x − 4 ) d x

9. Graphical Applications of Integrals

Area Between Curves

Business Calculus

9. Graphical Applications of Integrals

Area Between Curves

Struggling with Business Calculus?

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Multiple Choice

Shade the region bounded by $f (x) = l o g_{2} x + 4$ & $g (x) = e^{x - 4}$ on the interval [1,4]. Then set up an integral to represent the region's area.

Graph showing the area between the curves f(x) and g(x) shaded in yellow on the interval [1,4].

$A=\int_1^4\left(log_2x+4-e^{x-4}\right)dx$

Graph showing the area between the curves f(x) and g(x) shaded in yellow, bounded on the interval [1,4].

$A=\int_1^4\left(log_2x+4-e^{x-4}\right)dx$

Graph showing the area between the curves f(x) and g(x) shaded in yellow, with labeled axes and functions.

$A=\int_1^4\left(log_2x+4+e^{x-4}\right)dx$

Graph showing the area between the curves f(x) and g(x) shaded in yellow on the interval [1,4].

$A=\int_1^4\left(log_2x+4+e^{x-4}\right)dx$

Verified step by step guidance

Step 1: Identify the functions that bound the shaded region. The upper boundary is given by f(x) = log₂(x) + 4, and the lower boundary is given by g(x) = e^(x-4).

Step 2: Determine the interval over which the region is bounded. From the graph, the shaded region is bounded between x = 1 and x = 4.

Step 3: Set up the integral to calculate the area of the shaded region. The area is found by subtracting the lower function g(x) from the upper function f(x) over the interval [1, 4].

Step 4: Write the integral expression for the area: A = ∫₁⁴ [log₂(x) + 4 - e^(x-4)] dx.

Step 5: Ensure the integral is properly set up and ready for evaluation. The integrand represents the difference between the upper and lower functions, and the limits of integration correspond to the interval [1, 4].

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