The diagram shows the graphs of the functions $f(x)$ and $g(x)$. It is known that $$\int_{a}^{c} f(x) \, dx = 10$$ $$\int_{a}^{b} g(x) \, dx = -2$$ $$\int_{b}^{c} g(x) \, dx = 3.$$ What is the area between the curves $y = f(x)$ and $y = g(x)$ between $x = a$ and $x = c$?

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The diagram shows the graphs of the functions $f(x)$ and $g(x)$ - HSC - SSCE Mathematics Extension 1 - Question 4 - 2023 - Paper 1

Question 4

The diagram shows the graphs of the functions $f(x)$ and $g(x)$. It is known that $$\int_{a}^{c} f(x) \, dx = 10$$ $$\int_{a}^{b} g(x) \, dx = -2$$ $$\int_{b}... show full transcript

Worked Solution & Example Answer:The diagram shows the graphs of the functions $f(x)$ and $g(x)$ - HSC - SSCE Mathematics Extension 1 - Question 4 - 2023 - Paper 1

Step 1

Calculate the total area under $f(x)$ from $a$ to $c$

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Answer

The area under the curve $f(x)$ from $a$ to $c$ is given by: $A_f = \int_{a}^{c} f(x) \, dx = 10.$

Step 2

Calculate the total area under $g(x)$ from $a$ to $c$

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Answer

To find the area under $g(x)$ from $a$ to $c$ , we can add the areas from $a$ to $b$ and $b$ to $c$ :

The area from $a$ to $b$ is $\int_{a}^{b} g(x) \, dx = -2.$
The area from $b$ to $c$ is $\int_{b}^{c} g(x) \, dx = 3.$

Thus, the total area under $g(x)$ from $a$ to $c$ is: $A_g = \int_{a}^{b} g(x) \, dx + \int_{b}^{c} g(x) \, dx = -2 + 3 = 1.$

Step 3

Determine the area between the curves

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Answer

The area between the curves from $x = a$ to $x = c$ is calculated as:

$\text{Area} = A_f - A_g = 10 - 1 = 9.$

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