MHB Find Area Shaded in Red of Curve f(x)

  • Thread starter Thread starter MarkFL
  • Start date Start date
  • Tags Tags
    Area
AI Thread Summary
The area shaded in red under the curve f(x) = (e^x + x)(e^x + 1) is calculated using the integral A = ∫[x0, 0] f(x) dx - 2/5, where x0 is the point where f(x0) = 0. The integral evaluates to 1/2, leading to the final area A being 1/10 after subtracting the area of the triangle. The discussion emphasizes the importance of accurately determining x0 and evaluating the integral. The method used by participants confirms the correctness of the calculations.
MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Consider the following diagram:

View attachment 1248

The curve is given by:

$$f(x)=\left(e^x+x \right)\left(e^x+1 \right)$$

The tangent line is tangent to the curve at $x=0$.

Please compute the area shaded in red.
 

Attachments

  • areabetweencurveandtangent.jpg
    areabetweencurveandtangent.jpg
    7.1 KB · Views: 91
Mathematics news on Phys.org
[sp]Let $x_0$ be the point such that $f(x_0) = 0$, so that $e^{x_0} + x_0 = 0$ (because the other factor in $f(x)$ is always positive). The shaded area is given by $$A = \int_{x_0}^0f(x)\,dx - \tfrac25,$$ the $\frac25$ being the area of the triangle comprising the unshaded area under the curve. So we have to evaluate $$\int_{x_0}^0f(x)\,dx = \int_{x_0}^0(e^{2x} + (x+1)e^x + x)\,dx = \Bigl[\tfrac12e^{2x} + e^x + \tfrac12x^2\Bigl]_{x_0}^0 = \tfrac12\Bigl[( e^x + x)^2\Bigl]_{x_0}^0 = \tfrac12(1-0) = \tfrac12.$$ Thus $A = \frac12 - \frac25 = \frac1{10}.$[/sp]
 
Hello Opalg,

Spot on as always. (Sun)

I used essentially the same method:

This problem was originally posted about a year ago on another forum, and the student was confused by not having a lower limit of integration. He knew zero was the upper limit, but was unable to determine the root of the curve.

He had already determined the equation of the tangent line to be:

$$y=5x+2$$

To see that this is true, we find by differentiating $f$ with respect to $x$:

$$f'(x)=\left(e^x+x \right)e^x+\left(e^x+1 \right)^2$$

Hence, we find:

$$f'(0)=5$$

and then applying the point-slope formula, we have:

$$y-2=5(x-0)$$

$$y=5x+2$$

And so the area under the tangent line and bounded by the axes is a right triangle whose base is the magnitude of the $x$-intercept, or $\dfrac{2}{5}$, and whose altitude is $2$, which means the area under the line is:

$$A_T=\frac{1}{2}\cdot\frac{2}{5}\cdot2=\frac{2}{5}$$

To find the area under the curve, I instructed him to let $x_0$ represent this root, and so we may write:

$$A_C=\int_{x_0}^0 \left(e^x+x \right)\left(e^x+1 \right)\,dx$$

Next, use the substitution:

$$u=e^x+x\,\therefore\,du=\left(e^x+1 \right)\,dx$$

Since $f\left(x_0 \right)=0$, then we must have $e^{x_0}+x_0=0$ as well, because the other factor $e^x+1$ has no real roots. And so our definite integral becomes:

$$A_C=\int_0^1 u\,du=\frac{1}{2}\left[u^2 \right]_0^1=\frac{1}{2}$$

Finally, since the shaded area $A$ is the area under the curve less the area under the tangent line, we may write:

$$A=A_C-A_T=\frac{1}{2}-\frac{2}{5}=\frac{1}{10}$$
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

Similar threads

Back
Top