The area (in sq. units) bounded by `y=2^(x) and y=2x-x^(2)` from x = 1 to x = 2 is `k log_(2)e-l`, then the value of `|(k)/(l)|` is equal to

Answer

To find the area bounded by the curves \( y = 2^x \) and \( y = 2x - x^2 \) from \( x = 1 \) to \( x = 2 \), we will follow these steps:

### Step 1: Set up the area integral
The area \( A \) between the two curves from \( x = 1 \) to \( x = 2 \) can be found using the formula:
\[ A = \int_{1}^{2} (f(x) - g(x)) \, dx \]
where \( f(x) = 2^x \) (the upper curve) and \( g(x) = 2x - x^2 \) (the lower curve).

### Step 2: Compute the integral of \( f(x) = 2^x \)
We need to calculate:
\[ \int_{1}^{2} 2^x \, dx \]
The integral of \( 2^x \) is given by:
\[ \int 2^x \, dx = \frac{2^x}{\ln(2)} \]
Thus,
\[ \int_{1}^{2} 2^x \, dx = \left[ \frac{2^x}{\ln(2)} \right]_{1}^{2} = \frac{2^2}{\ln(2)} - \frac{2^1}{\ln(2)} = \frac{4}{\ln(2)} - \frac{2}{\ln(2)} = \frac{2}{\ln(2)} \]

### Step 3: Compute the integral of \( g(x) = 2x - x^2 \)
Next, we compute:
\[ \int_{1}^{2} (2x - x^2) \, dx \]
The integral is:
\[ \int (2x - x^2) \, dx = x^2 - \frac{x^3}{3} \]
Thus,
\[ \int_{1}^{2} (2x - x^2) \, dx = \left[ x^2 - \frac{x^3}{3} \right]_{1}^{2} = \left( 2^2 - \frac{2^3}{3} \right) - \left( 1^2 - \frac{1^3}{3} \right) \]
Calculating this gives:
\[ = \left( 4 - \frac{8}{3} \right) - \left( 1 - \frac{1}{3} \right) = \left( 4 - \frac{8}{3} \right) - \left( \frac{3}{3} - \frac{1}{3} \right) = \left( 4 - \frac{8}{3} \right) - \frac{2}{3} \]
Converting \( 4 \) to a fraction:
\[ = \left( \frac{12}{3} - \frac{8}{3} \right) - \frac{2}{3} = \frac{12 - 8 - 2}{3} = \frac{2}{3} \]

### Step 4: Calculate the area
Now we can find the area:
\[ A = \int_{1}^{2} (2^x - (2x - x^2)) \, dx = \frac{2}{\ln(2)} - \frac{2}{3} \]

### Step 5: Simplify the area expression
To express the area in the form \( k \log_2 e - l \), we need to manipulate the expression:
\[ A = \frac{2}{\ln(2)} - \frac{2}{3} \]
Using the change of base formula \( \ln(2) = \log_2 e \), we rewrite:
\[ A = 2 \log_2 e - \frac{2}{3} \]
Here, we can identify \( k = 2 \) and \( l = \frac{2}{3} \).

### Step 6: Calculate \( \left| \frac{k}{l} \right| \)
Now we find:
\[ \left| \frac{k}{l} \right| = \left| \frac{2}{\frac{2}{3}} \right| = \left| 2 \cdot \frac{3}{2} \right| = 3 \]

### Final Answer
Thus, the value of \( \left| \frac{k}{l} \right| \) is \( 3 \).

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