Question

Area bounded by the curves y = `"e"^(x^2)`, the x-axis and the lines x = 1, x = 2 is given to be α square units. If the area bounded by the curve y = `sqrt(ℓ "n"x)`, the x-axis and the lines x = e and x = e⁴ is expressed as (pe⁴ – qe – α), (where p and q are positive integers), then (p + q) is ______.

विकल्प

• 0.00

• 1.00

• 2.00

• 3.00

Answer 1

Area bounded by the curves y = `"e"^(x^2)`, the x-axis and the lines x = 1, x = 2 is given to be α square units. If the area bounded by the curve y = `sqrt(ℓ "n"x)`, the x-axis and the lines x = e and x = e⁴ is expressed as (pe⁴ – qe – α), (where p and q are positive integers), then (p + q) is 3.00.

Explanation:

`int_1^2"e"^(x^2)"d"x` = α

`int_"e"^("e"^4) sqrt(ℓ "n"x)"d"x` = pe⁴ – qe – α

`int_1^2"e"^(x^2)"d"x + int_"e"^("e"^4) sqrt(ℓ nx)"d"x` = pe⁴ – qe

⇒ 2e⁴ – e = pe⁴ – qe

⇒ p = 2, q = 1

⇒ p + q = 3      ...`[(∵ int_alpha^beta"f"(x)"d"x + int_"a"^"b" "f"^-1(x)"d"x="b" beta -"a"alpha),("when"  "a" = "f" (alpha)  &   "b" = "f"(beta))]`