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Oduvanchick [21]
2 years ago
13

Evaluate the definite integral using the graph of f(x) (Image included)

Mathematics
1 answer:
Tanya [424]2 years ago
8 0

a) The first integral corresponds to the area under y = f(x) on the interval [0, 3], which is a right triangle with base 3 and height 5, hence the integral is

\displaystyle \int_0^3 f(x) \, dx = \frac12 \times 3 \times 5 = \boxed{\frac{15}2}

b) The integral is zero since the areas under the curve over [3, 4] and [4, 5] are equal but opposite in sign. In other words, on the interval [3, 5], f(x) is symmetric and odd about x = 4, so

\displaystyle \int_3^5 f(x) \, dx = \int_3^4 f(x) \, dx + \int_4^5 f(x) \, dx = \int_3^4 f(x) \, dx - \int_3^4 f(x) \, dx = \boxed{0}

c) The integral over [5, 9] is the negative of the area of a rectangle with length 9 - 5 = 4 and height 5, so

\displaystyle \int_5^9 f(x) \, dx = -4\times5 = -20

Then by linearity, we have

\displaystyle \int_0^9 f(x) \, dx = \left\{\int_0^3 + \int_3^5 + \int_5^9\right\} f(x) \, dx = \frac{15}2 + 0 - 20 = \boxed{-\frac{25}2}

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Answer:

x = 14

Step-by-step explanation:

Since the triangles are similar then corresponding sides are in proportion, that is

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12(x - 4) = 120 ( divide both sides by 12 )

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Answer:

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Step-by-step explanation:

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Allyson and Adrian have decided to connect their ankles with a bungie cord; one end is tied to each person's ankle. The cord is
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3 years ago
Farmer Jonson is putting together vegetable baskets. He has 270 carrots and 180 beets. What is the largest number of baskets he
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Factors of 180:

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.


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What are the factors of both 270 and 180?

1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.

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Answer: 90 baskets.

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