Use a deﬁnite integral to ﬁnd an expression that represents the area of the region between the given curve and the
x-axis on the interval [0, b].
-y = 10x
A) $- 5 b ^ { 2 }$
B) $5 b ^ { 2 }$
C) $10 \mathrm {~b}$
D) $- 10 b$

Question

Use a deﬁnite integral to ﬁnd an expression that represents the area of the region between the given curve and the
x-axis on the interval [0, b].
-y = 10x 
A) $- 5 b ^ { 2 }$
B) $5 b ^ { 2 }$
C) $10 \mathrm {~b}$
D) $- 10 b$

Quizplus · Accepted Answer

To find the area between the curve and the x-axis, we need to integrate the equation of the curve with respect to x over the interval [0, b]:

\begin{aligned} A &=\int_{0}^{b} 10 x d x \ &=\left.\frac{10 x^{2}}{2}ight|_{0}^{b} \ &=\frac{10 b^{2}}{2} \ &=5 b^{2} \end{aligned}

Therefore, the expression that represents the area of the region between the given curve and the x-axis on the interval [0, b] is 5b^2, which corresponds to choice B.

Use a Deﬁnite Integral to ﬁNd an Expression That Represents −5b2- 5 b ^ { 2 }−5b2

Use a Deﬁnite Integral to ﬁNd an Expression That Represents $- 5 b ^ { 2 }$