Area Between Curves Practice Problems
Problem: Area Between Curves #1
Find the area between the curve:
\Large{y =x^2-2x}, and the x-axis
for the interval: [2, 4]
We are being asked to find the area under the given function for the specified intervals. Note that since we are computing the area between the function and the x axis, all we need to do is computer the definite integral as we normally would.
Step 1:
The definite integral is as follows:
\large{\int_{2}^{4}(x^2-2x)dx = (\frac{1}{3}x^3 – x^2)|_{2}^{4}}
Step 2:
Now we can apply the fundamental theorem of calculus to evaluate. This states that:
\large{\int_{a}^{b}f(x) dx = F(b) – F(a)}
Applying the fundamental theorem of calculus, we have:
\large{\int_{2}^{4}(x^2-2x) dx = [\frac{1}{3}(4)^3 – (4)^2] – [\frac{1}{3}(2)^3 – (2)^2]}
\large{ \approx 5.33 – (- 1.33)}
\large{\approx 6.67}
Thus, the area between the function y = x^2-2x and the x axis is approximately 6.67 units.
Solution Completed!
We are being asked to find the area under the given function for the specified intervals. Note that since we are computing the area between the function and the x axis, all we need to do is computer the definite integral as we normally would.
Step 1:
The definite integral is as follows:
\large{\int_{2}^{4}(x^2-2x)dx = (\frac{1}{3}x^3 – x^2)|_{2}^{4}}
Step 2:
Now we can apply the fundamental theorem of calculus to evaluate. This states that:
\large{\int_{a}^{b}f(x) dx = F(b) – F(a)}
Applying the fundamental theorem of calculus, we have:
\large{\int_{2}^{4}(x^2-2x) dx}
\large{ = [\frac{1}{3}(4)^3 – (4)^2] – [\frac{1}{3}(2)^3 – (2)^2]}
\large{ \approx 5.33 – (- 1.33)}
\large{\approx 6.67}
Thus, the area between the function y = x^2-2x and the x axis is approximately 6.67 units.
Solution Completed!
Problem: Area Between Curves #2
Find the area of the region between:
\Large{y = sin(x)}, and \Large{y = cos(x)}
for the interval: [0, \frac{\pi}{4}]
\Large{y = sin(x)}, and \Large{y = cos(x)}
for the interval: [0, \frac{\pi}{4}]
Problem: Area Between Curves #3
Find the area of the region bounded by:
\Large{y = e^{-\frac{1}{2}x}}
and
\Large{y = x\sqrt{x^2+1}}
for the interval: [-3, 0]
Problem: Area Between Curves #4
Find the area of the region between the functions:
\large{f(x) = x^2} and \large{g(x) = 8-x^2}
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