Evaluate the integral by interpreting it in terms of areas: $6{\int }_{-5}^0\left|x\right| dx$.

• A. $60$
• B. $50$
• C. $75$
• D. $25$

Answer: (C)

Verified step by step guidance

1. Step 1: Recognize that the integral involves the absolute value function |x|, which is defined as |x| = x if x ≥ 0 and |x| = -x if x < 0. Since the interval of integration is [-5, 0], the function |x| simplifies to -x because x is negative in this range.
2. Step 2: Rewrite the integral using the simplified form of |x|. The integral becomes 6 ∫_{-5}^{0} (-x) dx.
3. Step 3: Factor out the constant 6 from the integral. This gives 6 * ∫_{-5}^{0} (-x) dx.
4. Step 4: Compute the integral ∫_{-5}^{0} (-x) dx. To do this, find the antiderivative of -x, which is (-x^2)/2, and evaluate it at the bounds -5 and 0.
5. Step 5: Interpret the result of the integral as the area under the curve. Since the absolute value function represents a geometric shape (a triangle in this case), calculate the area of the triangle formed by the graph of |x| over the interval [-5, 0] and multiply it by 6.