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Find an unknown constant in a curve from its given area

This type of question, asked 4 times: 2021 Paper 5, 26 Jul 2022, 6 Apr 2024, 23 Jan 2025. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 5

    88
    Let the curve y=y(x)y=y(x) be the solution of the differential equation, dydx=2(x+1)\frac{dy}{dx}= 2(x+ 1). If the numerical value of area bounded by the curve y=y(x)y=y(x) and xx-axis is 483\frac{4\sqrt{8}}{3}, then the value of y(1)y(1) is equal to

  2. Question 2

    26 Jul 2022

    65
    The odd natural number a, such that the area of the region bounded by y=1,y=3,x=0,x=yay = 1,y = 3,x = 0,x =y^{a} is 3643\frac{364}{3}, is equal to:

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  3. Question 3

    6 Apr 2024

    168
    If the area of the region {(x,y):ax2≤y≤1x,1≤x≤2,0<a<1}\left\{ (x,y):\frac{a}{x^{2}}\leq y \leq\frac{1}{x},1 \leq x \leq 2,0 < a < 1 \right\} is (log⁡e2)−17\left( \log_{e}2 \right)-\frac{1}{7} then the value of 7a−37a - 3 is equal to:

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  4. Question 4

    23 Jan 2025

    133
    If the area of the region {(x,y):−1≤x≤1,0≤y≤a+e∣x∣−e−x,a>0}\left\{ (x,y): - 1 \leq x \leq 1,0 \leq y \leq a +e^{|x|}-e^{- x},a > 0 \right\} is e2+8e+1e\frac{e^{2}+ 8e + 1}{e}, then the value of a is :

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