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Find the specified area under the graph of \( f \) over the interval \((-1, 3)\).

Given:
\[ f(x) = 5, \text{ if } x < 1 \]
\[ f(x) = 5x^2, \text{ if } x \geq 1 \]

Options:
A. 140
B. 20
C. 54
D. 160

Answer :

To find the total area under the graph of f over the interval (-1,3),

we add the two areas: 10 + 130/3 = 160/3.

Therefore, the answer is D) 160.

To find the area under the graph of f over the interval (-1,3)

for the function f(x) = {5, if x < 1 and 5x^2 if x>=1},

we need to calculate the area of the two parts separately since the function is piecewise.

Let's first find the area under the graph of f over the interval (-1,1):

For x < 1, the function f(x) = 5, which is a horizontal line.

Therefore, the area under the graph of f(x) over the interval (-1,1) is simply the area of the rectangle with height 5 and width 2 (since the interval is from -1 to 1).

This gives us an area of 5 * 2 = 10.

Now, let's find the area under the graph of f over the interval (1,3):

For x >= 1, the function f(x) = 5x^2, which is a parabola.

Therefore, we need to find the area under this curve over the interval (1,3).

The antiderivative of 5x^2 is (5/3)x^3, so we can use the definite integral to find the area:

∫[1,3] 5x^2 dx

= [(5/3)x^3]1,3

= (5/3)(3^3 - 1^3)

= (5/3)(26)

= 130/3

So the area under the graph of f over the interval (1,3) is 130/3.

To find the total area under the graph of f over the interval (-1,3),

we add the two areas:

10 + 130/3 = 160/3.

Therefore, the answer is D) 160.

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