23 uždavinys
Figūra yra ribojama parabolės y = x2 + 1 ir tiesės y = ax + 1; čia a > 0. Su kuria a reikšme šios figūros plotas lygus 36 ?
Sprendimas:
Randame grafikų susikirtimo taškus:
x^2+1 =
a* x+1
a* x+1
|
x^2+1 = a* x+1$$x^{2}+1$$ = $$a\cdot x+1$$
x^2- a* x+1 = 1$$x^{2}-a\cdot x+1$$ = $$1$$
x^2- a* x-1+1 = 0$$x^{2}-a\cdot x-1+1$$ = $$0$$
x^2- a* x+0 = 0$$x^{2}-a\cdot x+0$$ = $$0$$
x^2- a* x = 0$$x^{2}-a\cdot x$$ = $$0$$
Paaiškinimas: x* (x-a) = 0$$x\cdot (x-a)$$ = $$0$$
x = 0$$x$$ = $$0$$
x-a = 0$$x-a$$ = $$0$$
x = 0+a$$x$$ = $$0+a$$
x = a$$x$$ = $$a$$
Grafikai kertasi taškuose x = 0 ir x = a.
Norint rasti figūros plotą, reikia rasti tiesės ir parabolės skirtumo integralą nuo 0 iki a.
∫(0;a; a* x+1-( x^2+1)) =
36
36
|
|
∫(0;a; a* x+1-( x^2+1)) = 36$$\int_{0}^{a} (a\cdot x+1-(x^{2}+1))$$ = $$36$$
∫(0;a; a* x+1- x^2-1) = 36$$\int_{0}^{a} (a\cdot x+1-x^{2}-1)$$ = $$36$$
∫(0;a; a* x- x^2+1-1) = 36$$\int_{0}^{a} (a\cdot x-x^{2}+1-1)$$ = $$36$$
∫(0;a; a* x- x^2+0) = 36$$\int_{0}^{a} (a\cdot x-x^{2}+0)$$ = $$36$$
∫(0;a; a* x- x^2) = 36$$\int_{0}^{a} (a\cdot x-x^{2})$$ = $$36$$
Paaiškinimas: (0;a;
* x^2-
* x^3) = 36$$(\frac{a}{2}\cdot x^{2}-\frac{1}{3}\cdot x^{3}){\LARGE |}_{0}^{a}$$ = $$36$$
| a |
| / 2 |
| 1 |
| / 3 |
Paaiškinimas: (
* a^2-
* a^3)-(
* 0^2-
* 0^3) = 36$$(\frac{a}{2}\cdot a^{2}-\frac{1}{3}\cdot a^{3})-(\frac{a}{2}\cdot 0^{2}-\frac{1}{3}\cdot 0^{3})$$ = $$36$$
| a |
| / 2 |
| 1 |
| / 3 |
| a |
| / 2 |
| 1 |
| / 3 |
(
* a^2-
* a^3) = 36$$(\frac{a}{2}\cdot a^{2}-\frac{1}{3}\cdot a^{3})$$ = $$36$$
| a |
| / 2 |
| 1 |
| / 3 |
(
-
* a^3) = 36$$(\frac{a^{3}}{2}-\frac{1}{3}\cdot a^{3})$$ = $$36$$
| a^3 |
| / 2 |
| 1 |
| / 3 |
(
-
) = 36$$(\frac{a^{3}}{2}-\frac{a^{3}}{3})$$ = $$36$$
| a^3 |
| / 2 |
| a^3 |
| / 3 |
| a^3 |
| / 2 |
| a^3 |
| / 3 |
| a^3 |
| / 6 |
| a^3 |
| / 6 |
a^3 = 6* 6* 6$$a^{3}$$ = $$6\cdot 6\cdot 6$$
saknis(3, a^3) = saknis(3, 6* 6* 6)$$\sqrt[3]{a^{3}}$$ = $$\sqrt[3]{6\cdot 6\cdot 6}$$
a^3) = saknis(3, 6* 6* 6)$$\sqrt[3]{a^{3}}$$ = $$\sqrt[3]{6\cdot 6\cdot 6}$$a = saknis(3, 6* 6* 6)$$a$$ = $$\sqrt[3]{6\cdot 6\cdot 6}$$
6* 6* 6)$$a$$ = $$\sqrt[3]{6\cdot 6\cdot 6}$$a = 6$$a$$ = $$6$$
Atsakymas: a = 6