#f(x)=4x^2-25#
Find roots:
#4x^2-25=0#
#x=sqrt(25/4)=>x=5/2 and -5/2#
So our given area lies below and above the x axis.
It is below in the interval #[-1,5/2]#
and above in the interval #[5/2, 7/2]#
So we require the integrals:
#int_(-1)^(5/2)(4x^2-25)dxcolor(white)(88)# and #color(white)(88)int_(5/2)^(7/2)(4x^2-25)dx#
1st:
#"Area" =int_(-1)^(5/2)(4x^2-25)dx=4/3x^3-25x=[4/3x^3-25x]_(-1)^(5/2)#
#=[4/3x^3-25x]^(5/2)-[4/3x^3-25x]_(-1)#
Plugging in upper and lower bounds:
#=[4/3(5/2)^3-25(5/2)]^(5/2)-[4/3(-1)^3-25(-1)]_(-1)#
#=[125/6-125/2]^(5/2)-[-4/3+25]_(-1)#
#=[-125/3]^(5/2)-[71/3]_(-1)#
#=-125/3-71/3=-196/3#
#"Area" = 196/3#
2nd:
#"Area" =int_(5/2)^(7/2)(4x^2-25)dx=4/3x^3-25x#
#=[4/3x^3-25x]_(5/2)^(7/2)#
#=[4/3x^3-25x]^(7/2)-[4/3x^3-25x]_(5/2)#
Plugging in upper and lower bounds:
#=[4/3(7/2)^3-25(7/2)]^(7/2)-[4/3(5/2)^3-25(5/2)]_(5/2)#
#=[343/6-175/2]^(7/2)-[125/6-125/2]_(5/2)#
#=-91/3-(-125/3)=34/3#
#"Area = 34/3#
Total area #= 196/3+34/3=230/3=76.67# ( 2 .d.p.)
GRAPH:
You probably forgot to remove the negation when adding the two areas.
