Use geometry to find the exact value of $\int _ { - 1 } | 2 x + 1 | d x$
A) 20.25
B) 20
C) 20.5
D) 19.75

Question

Quizplus · Accepted Answer

The integral $\int _ { - 1 } | 2 x + 1 | d x$ can be interpreted geometrically as the area under the curve of $|2x + 1|$ from $x = -1$ to $x = 0$. The function $2x + 1$ crosses the x-axis at $x = -0.5$, dividing the interval into two parts. From $x = -1$ to $x = -0.5$, the function is negative, and its absolute value is represented by the line $2x + 1$ reflected above the x-axis, forming a triangle with base $0.5$ and height $1$. The area of this triangle is $\frac{1}{2} \times base \times height = \frac{1}{2} \times 0.5 \times 1 = 0.25$. From $x = -0.5$ to $x = 0$, the function $2x + 1$ is positive, and its graph is a triangle with base $0.5$ and height $1$, giving the same area of $0.25$. The total area, which is the value of the integral, is the sum of these two areas, $0.25 + 0.25 = 0.5$.

Use Geometry to Find the Exact Value Of ∫−1∣2x+1∣dx\int _ { - 1 } | 2 x + 1 | d x∫−1​∣2x+1∣dx

Use Geometry to Find the Exact Value Of $\int _ { - 1 } | 2 x + 1 | d x$