Find the area bounded by the following functions:
[tex]
\begin{array}{c}
y=\sqrt{x} \
y=\sqrt{8-x} \
y=0 \
\text { Area }=[?] \
\end{array}
[/tex]
Round your answer to the nearest thousandth.
Answer (2)
Find the intersection points of the curves y = x , y = 8 − x , and y = 0 .
Set up the integrals to find the area: A = ∫ 0 4 x d x + ∫ 4 8 8 − x d x .
Evaluate the integrals: ∫ 0 4 x d x = 3 16 and ∫ 4 8 8 − x d x = 3 16 .
Calculate the total area: A = 3 16 + 3 16 = 3 32 ≈ 10.667 .
Explanation
Problem Setup We are asked to find the area bounded by the curves y = x , y = 8 − x , and y = 0 . To find this area, we need to determine the points of intersection between these curves.
Finding Intersection Points First, let's find the intersection points:
Intersection of y = x and y = 0 : Setting x = 0 , we get x = 0 . So the intersection point is ( 0 , 0 ) .
Intersection of y = 8 − x and y = 0 : Setting 8 − x = 0 , we get 8 − x = 0 , so x = 8 . The intersection point is ( 8 , 0 ) .
Intersection of y = x and y = 8 − x : Setting x = 8 − x , we get x = 8 − x , which gives 2 x = 8 , so x = 4 . Then y = 4 = 2 . The intersection point is ( 4 , 2 ) .
Setting up the Integrals Now we can set up the integrals to find the area. The area is bounded by the x-axis ( y = 0 ) and the two curves y = x and y = 8 − x . We can split the area into two regions:
From x = 0 to x = 4 , the area is given by the integral of y = x .
From x = 4 to x = 8 , the area is given by the integral of y = 8 − x .
So the total area is the sum of these two integrals: A = ∫ 0 4 x d x + ∫ 4 8 8 − x d x
Evaluating the First Integral Let's evaluate the first integral: ∫ 0 4 x d x = ∫ 0 4 x 1/2 d x = 3 2 x 3/2 0 4 = 3 2 ( 4 3/2 − 0 3/2 ) = 3 2 ( 8 − 0 ) = 3 16
Evaluating the Second Integral Now let's evaluate the second integral: ∫ 4 8 8 − x d x Let u = 8 − x , so d u = − d x . When x = 4 , u = 4 . When x = 8 , u = 0 . Thus, ∫ 4 8 8 − x d x = − ∫ 4 0 u d u = ∫ 0 4 u d u = 3 2 u 3/2 0 4 = 3 2 ( 4 3/2 − 0 3/2 ) = 3 2 ( 8 − 0 ) = 3 16
Calculating the Total Area The total area is the sum of the two integrals: A = 3 16 + 3 16 = 3 32
Final Answer The total area is 3 32 , which is approximately 10.666666... . Rounding to the nearest thousandth, we get 10.667 .
Examples
Imagine you are designing a solar panel where the sunlight is focused onto a collector. The shape of the collector is defined by the curves y = x and y = 8 − x . To optimize the panel's efficiency, you need to calculate the area between these curves, which represents the surface area available for collecting sunlight. This problem demonstrates how finding the area between curves can be applied in engineering design to maximize performance.
The area bounded by the curves y = x , y = 8 − x , and the x-axis is calculated by finding intersection points and setting up integrals. The final area is 3 32 or approximately 10.667 when rounded to the nearest thousandth.
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