The graph of function $h$ follows. Let $f(x)=\int_{-1}^{x} h(t) d t$ . $\newline$ Evaluate $f(4)$ . $\newline$ $f(4)=$

Full solution

Q. The graph of function

h

follows. Let

f(x)=\int_{-1}^{x} h(t) d t

\newline

Evaluate

f(4)

\newline

f(4)=

Substitute $x$ with $4$ : Substitute $x$ with $4$ in the function $f(x)$ . $f(4) = \int_{-1}^{4} h(t) \, dt$
Graph of $h(t)$ : Look at the graph of $h(t)$ to find the area under the curve from $-1$ to $4$ . The graph shows a triangle from $-1$ to $0$ and a rectangle from $0$ to $4$ .
Calculate triangle area: Calculate the area of the triangle. $\newline$ Area of triangle = $\frac{1}{2} \times \text{base} \times \text{height}$ $\newline$ Base = $1$ (from $-1$ to $0$ ), height = $2$ (from the x-axis to the peak of the triangle) $\newline$ Area\_triangle = $\frac{1}{2} \times 1 \times 2 = 1$
Calculate rectangle area: Calculate the area of the rectangle. $\newline$ Area of rectangle = $\text{length} \times \text{width}$ $\newline$ Length = $4$ (from $0$ to $4$ ), width = $2$ (from the $x$ -axis to the top of the rectangle) $\newline$ Area\_rectangle = $4 \times 2 = 8$
Find total area: Add the areas of the triangle and rectangle to find the total area under the curve. $\newline$ Total area = $\text{Area}_{\text{triangle}} + \text{Area}_{\text{rectangle}}$ $\newline$ Total area = $1 + 8 = 9$
Total area under the curve: The total area under the curve from $-1$ to $4$ is the value of $f(4)$ . $\newline$ $f(4) = 9$