Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { 1 } ^ { 4 } 2 s d s$
A) 30
B) 15
C) 4
D) 34
E)-17

Question

Quizplus · Accepted Answer

To sketch the region, we start with the equation of the line $y = 2x$ and the x-axis. We then find the $x$ values where the line intersects the x-axis: $$0 = 2x\Rightarrow x=0$$ We need to integrate from $x=1$ to $x=4$, so we start by evaluating $f(1)$ and $f(4)$: $$f(1) = 2(1) = 2$$ $$f(4) = 2(4) = 8$$ The region we need to find the area of is shown below:

[asy]
unitsize(0.6cm);
pair A = (1,2);
pair B = (4,8);
draw (A--B,linewidth(1));
draw (A--(B.x,0),linewidth(1));
fill((1,2)--(4,8)--(4,0)--(1,0)--cycle, gray(0.7));
draw((0,-1)--(0,10),Arrows(0.15cm));
draw((-1,0)--(7,0),Arrows(0.15cm));
Label f;

f.p=fontsize(3);

xaxis(-1, 7, Ticks(f, 1.0));

yaxis(-1, 10, Ticks(f, 1.0));

label("$y=2x$",(5,9), N,fontsize(10));
label("$x$",(7,0),SE,fontsize(10));
label("$y$",(0,10),N,fontsize(10));
label("$(1,2)$",(1,2),NW,fontsize(10));
label("$(4,8)$",(4,8),NW,fontsize(10));
[/asy]

We can now use the formula for the area of a trapezoid: $$	ext{Area}=\frac { a+b }{ 2 } h$$ where $a$ and $b$ are the lengths of the parallel sides and $h$ is the height. In this case, $a = 2$, $b = 8$, and $h = 3$, so: $$	ext{Area}=\frac { 2+8 }{ 2 } (3)=15$$ Therefore, the answer is $\boxed{	ext{(B)}}$.

Sketch the Region Whose Area Is Given by the Definite ∫142sds\int _ { 1 } ^ { 4 } 2 s d s∫14​2sds

Sketch the Region Whose Area Is Given by the Definite $\int _ { 1 } ^ { 4 } 2 s d s$