Question

Compute the area of the figure bounded by the straight lines x = 0, x = 2 and the curves y = 2^x, y = 2x – x² -

• A. $\frac { 3 } { \log 2 }$ – $\frac { 4 } { 3 }$
• B. $\frac { 3 } { \log 2 }$ + $\frac { 4 } { 3 }$
• C. $\frac { 3 } { \log 2 }$ –$\frac { 2 } { 3 }$
• D. None

Answer 1

Answer: (A)

$\frac { 3 } { \log 2 }$ – $\frac { 4 } { 3 }$

Answer 2

Figure is self explanatory. Given curves are

y = 2^x ...(i)

(x – 1)² = – (y – 1) ...(ii)

The required area

= dx

Where y₁ = 2x and y₂ = 2x – x²

= = $\left[ \frac { 2 } { \log 2 } - x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } \right] _ { 0 } ^ { 2 }$

= $\left( \frac { 4 } { \log 2 } - 4 + \frac { 8 } { 3 } \right)$ – $\frac { 1 } { \log 2 }$ = $\frac { 3 } { \log 2 }$ – $\frac { 4 } { 3 }$