1.
Find the positive value of c such that the area enclosed by the graph of y = -x^2 +c^2 and the x-axis is 36 .
2.
Let \[f(t) = \int_0^t (x - 7)(x - 4)(x + 1)(e^x - 1) \left( \arctan (x) - \frac{\pi}{3} \right) \: dx.\]
Find all values of t where f(t) has a local minimum at t
Find the positive value of c such that the area enclosed by the graph of y = -x^2 +c^2 and the x-axis is 36 .
Hello Guest!
\(f(x)=y=-x^2+c^2=0\\ x^2=c^2\\ \color{blue}x=\pm\ c\)
\(\int _{-c}^{+c}(-x^2+c^2)dx=\color{blue}|c^2x-\frac{x^3}{3}|_{-c}^{+c}=36\)
\(|c^2x-\frac{x^3}{3} |_{-c}^{+c}= (c^2\cdot c-\frac{c^3}{3}) - (c^2\cdot (-c)-\frac{(-c)^3}{3}) =36\)
\(c^3-\frac{c^3}{3}-(-c^3-\frac{(-c)^3}{3})=36\\ c^3-\frac{c^3}{3}+c^3-\frac{c^3}{3}=36\\ 3c^3-c^3+3c^3-c^3=3\cdot 36\\ 4c^3=108\\ c=\sqrt[3]{\frac{108}{4}}=\sqrt[3]{27}\)
\(c=3\)
!
Let
\(f(t) = \int_0^t (x - 7)(x - 4)(x + 1)(e^x - 1) \left( \arctan (x) - \frac{\pi}{3} \right) \: dx \) .
Find all values of t where f(t) has a local minimum at t.
Hello Guest!
All values of t, where \(f(x) = (x - 7)(x - 4)(x + 1)(e^x - 1) \left( \arctan (x) - \frac{\pi}{3} \right) \: \)
has a local minimum at x ar, ar \(x\in \{0,\ 5.712\}\)
Determined graphically.
!