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^c g... 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 function $f(x)$ from $a$ to $c$ is given by the integral:
[ A_f = \int_a^c f(x) , dx = 10 ]

Step 2

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

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Answer

The area under the function $g(x)$ from $a$ to $b$ is:
[ A_g(a, b) = \int_a^b g(x) , dx = -2 ]

The area under the function $g(x)$ from $b$ to $c$ is:
[ A_g(b, c) = \int_b^c g(x) , dx = 3 ]

Step 3

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

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Answer

The total area under $g(x)$ from $a$ to $c$ can be combined:
[ A_g(a, c) = A_g(a, b) + A_g(b, c) = -2 + 3 = 1 ]

Step 4

Calculate the area between the curves

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Answer

The area between the curves $y = f(x)$ and $y = g(x)$ from $x = a$ to $x = c$ is given by:
[ \text{Area} = A_f - A_g(a, c) = 10 - 1 = 9 ]

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