Question

Determine the area of the given curves

1 . 𝑦² = 2 𝑥 ; 𝑦 = 𝑥 − 4

2. 𝑦 = 𝑥 − 𝑥² ; 𝑦 = − 𝑥

Answer 1

1)  𝑦² = 2 𝑥 ; 𝑦 = 𝑥 − 4

S=$\int$_-2⁴(y+4-y²/2)dy=(y²/2+4y-y³/6)$\mid$_-2⁴=8+16-64/6-2+8-8/6=30-32/3-4/3=30-36/3=30-12=18

2) 𝑦 = 𝑥 − 𝑥² ; 𝑦 = − 𝑥

S=$\int$₀²(x-x²+x)dx=$\int$₀²(2x-x²) dx=(x²-x³/3)$\mid$₀²=4-8/3=4/3