Question

Find the area bounded by the X-axis the parabola y=-x²+4.

Answer 1

graph of y = -x² + 4 , is a parabolic as shown in figure. shaded region shows graph of bounded area enclosed by x -axis.

so, area of bounded region enclosed by x-axis = |$\int\limits^2_{-2}{y}\,dx$|

= |$\int\limits^2_{-2}{(-x^2+4)}\,dx$|

= |$\int\limits^2_{-2}{-x^2}\,dx+\int\limits^2_{-2}{4}\,dx$|

=| $-\left[\frac{x^3}{3}\right]^2_{-2}+4[x]^2_{-2}$|

= |-{2³ - (-2)³}/3 + 4[2 + 2] |

= |-16/3 + 16|

= 32/3 sq unit.

hence, the area bounded by the X-axis the parabola y=-x²+4, is 32/3 sq unit