Title: JSX Graphs
Course: HELM Workbooks 2023-2024

Regions \(A\) and \(B\) are bounded by the graph of \(f\) and the \(x\)-axis, as shown below.

Region \(A\) has area [value] and region \(B\) has area [value].

Find the value of \(\displaystyle\int_[value]^[value]\left(f(x)+[value]\right)\,dx\)
Feedback: First, we can break the integral up: \[\begin{align*} \int_[value]^[value]\left(f(x)+[value]\right)\,dx &= \int_[value]^[value]f(x)\,dx+ \int_[value]^[value]\left([value]\right)\,dx \end{align*}\] From the facts given in the question, we know \(\displaystyle\int_[value]^[value]f(x)\,dx=[value]+[value]=[value]\). Next, we calculate the definite integral: \[ \begin{align*}\int_[value]^[value]\left([value]\right)\,dx &= \left[[value]\right]_[value]^[value] \\ &= [value] \end{align*}\] Thus \(\displaystyle\int_[value]^[value]\left(f(x)+[value]\right)\,dx=[value]+[value]=[value]\).

The diagram shows the graph of a parabola with turning point \(([value],[value])\), and the line with equation \(y=[value]\), both of which pass through the point \(([value],[value])\).

(a) What is the equation of the parabola?

    \(y=\)

(b) What is the value of \(k\)?

    \(k=\)

(c) Calculate the shaded area, enclosed between the parabola and the line.

    The shaded area is
Feedback: (a) From the turning point, the equation of the parabola must be of the form \(y=[value]\). Since it passes through \(([value],[value])\), we have [value], thus [value]. So the parabola has equation [value]. (b) The line \(y=[value]\) also passes through \(([value],[value])\), so substituting gives [value], from which we get \(k=[value]\). (c) To find the shaded area, we need the points of intersection of the line and the parabola, so we solve: \[\begin{align*}[value]&=[value] \\ [value]-([value]) &=0 \\ [value] &=0 \\ [value]&=0, \end{align*}\] from which we see that the curves intersect when \(x=[value]\) or \(x=[value]\). So, the shaded area is given by \[\begin{align*} \int_[value]^[value] \left(\text{upper}-\text{lower}\right)\,dx &= \int_[value]^[value] \left(([value])-([value])\right)\,dx \\ &= \int_[value]^[value] \left([value]\right)\,dx. \\ &= \left[[value]\right]_[value]^[value] \\ &= \left([value]\right)-\left([value]\right) \\ &= [value]. \end{align*}\]

1. Regions \(A\) and \(B\) are bounded by the graph of \(f\) and the \(x\)-axis, as shown below.
Region \(A\) has area [value] and region \(B\) has area [value].
Find the value of \(\displaystyle\int_[value]^[value]\left(f(x)+[value]\right)\,dx\)

2. Find the total area enclosed between the \(x\)-axis and the curve \(y=[value]\) between \(x=[value]\) and \(x=[value]\).
Feedback: First, we can break the integral up: \[\begin{align*} \int_[value]^[value]\left(f(x)+[value]\right)\,dx &= \int_[value]^[value]f(x)\,dx+ \int_[value]^[value]\left([value]\right)\,dx \end{align*}\] From the facts given in the question, we know \(\displaystyle\int_[value]^[value]f(x)\,dx=[value]+[value]=[value]\). Next, we calculate the definite integral: \[ \begin{align*}\int_[value]^[value]\left([value]\right)\,dx &= \left[[value]\right]_[value]^[value] \\ &= [value] \end{align*}\] Thus \(\displaystyle\int_[value]^[value]\left(f(x)+[value]\right)\,dx=[value]+[value]=[value]\).

Graph of sine function should appear here:

Graph of function should appear here:

\( A = \)