NIMCET 2007 — Mathematics PYQ

Step 1: Find the Point of Intersection

To find where the two curves intersect, equate the two equations:

$$x{e}^x=x{e}^{-x}$$

$$x{e}^x-x{e}^{-x}=0$$

$$x(e^x-e^{-x})=0$$

This gives two possibilities:

1. $x=0$

2. $e^x-e^{-x}=0⟹e^x=e^{-x}⟹e^{2x}=1⟹x=0$

Thus, the curves intersect at exactly one point, which is $x=0$. At $x=0$, $y=0\cdot e^0=0$.

Step 2: Determine the Upper and Lower Curve

For the region from $x=0$ to $x=1$:

• Since e^x > e^{-x} for all x > 0, multiplying by a positive value $x$ gives:

  xe^x > xe^{-x}

Therefore, $y=x{e}^x$ is the upper curve and $y=x{e}^{-x}$ is the lower curve in this interval.

Step 3: Set Up the Area Integral

The area $A$ bounded between the curves from $x=0$ to $x=1$ is given by:

$$A={\int }_0^1(y_{\text{upper}}-y_{\text{lower}})dx$$

$$A={\int }_0^1(x{e}^x-x{e}^{-x})dx$$

Step 4: Evaluate the Integrals Using Integration by Parts

We will integrate each term using the Integration by Parts formula: $\int udv=uv-\int vdu$

Part 1: $\int x{e}^{x}dx$

• Let $u=x⟹du=dx$

• Let $dv=e^{x}dx⟹v=e^x$

$$\int x{e}^{x}dx=x{e}^x-\int e^{x}dx=x{e}^x-e^x$$

Part 2: $\int x{e}^{-x}dx$

• Let $u=x⟹du=dx$

• Let $dv=e^{-x}dx⟹v=-e^{-x}$

$$\int x{e}^{-x}dx=x(-e^{-x})-\int (-e^{-x})dx=-x{e}^{-x}-e^{-x}$$

Step 5: Apply the Limits

Combine the individual antiderivatives:

$$\int (x{e}^x-x{e}^{-x})dx=(x{e}^x-e^x)-(-x{e}^{-x}-e^{-x})=x{e}^x-e^x+x{e}^{-x}+e^{-x}$$

Now evaluate this from $0$ to $1$:

$$A={\left[x{e}^x-e^x+x{e}^{-x}+e^{-x}\right]}_0^1$$

Upper Limit ($x=1$):

$$\left(1\cdot e^1-e^1+1\cdot e^{-1}+e^{-1}\right)=e-e+\frac{1}{e}+\frac{1}{e}=\frac{2}{e}$$

Lower Limit ($x=0$):

$$\left(0\cdot e^0-e^0+0\cdot e^0+e^0\right)=0-1+0+1=0$$

Subtract the lower limit value from the upper limit value:

$$A=\frac{2}{e}-0=\frac{2}{e}$$

Conclusion

Since the exact value is $\frac{2}{e}$ (which is approximately $0.736$), it matches none of the numerical choices given in options (a), (b), or (c).

Correct Option: (d) none