1. **Tentukan volume benda putar yang dibatasi oleh kurva** $y = x^2 + 1$ dan $y = x + 3$ diputar mengelilingi sumbu $x$ sejauh 360°.
2. **Cari titik potong kedua kurva untuk menentukan batas integral:**
$$x^2 + 1 = x + 3$$
$$x^2 - x - 2 = 0$$
$$ (x - 2)(x + 1) = 0$$
Jadi, batas integral adalah $x = -1$ sampai $x = 2$.
3. **Gunakan rumus volume benda putar dengan metode washer:**
$$V = \pi \int_{a}^{b} \left[ (y_\text{atas})^2 - (y_\text{bawah})^2 \right] dx$$
Di sini, $y_\text{atas} = x + 3$ dan $y_\text{bawah} = x^2 + 1$.
4. **Substitusi fungsi ke dalam integral:**
$$V = \pi \int_{-1}^{2} \left[ (x + 3)^2 - (x^2 + 1)^2 \right] dx$$
5. **Kembangkan kuadrat:**
$$(x + 3)^2 = x^2 + 6x + 9$$
$$(x^2 + 1)^2 = x^4 + 2x^2 + 1$$
6. **Masukkan ke integral:**
$$V = \pi \int_{-1}^{2} \left( x^2 + 6x + 9 - x^4 - 2x^2 - 1 \right) dx$$
$$= \pi \int_{-1}^{2} \left( -x^4 - x^2 + 6x + 8 \right) dx$$
7. **Integralkan setiap suku:**
$$\int -x^4 dx = -\frac{x^5}{5}$$
$$\int -x^2 dx = -\frac{x^3}{3}$$
$$\int 6x dx = 3x^2$$
$$\int 8 dx = 8x$$
8. **Tuliskan integral lengkap:**
$$V = \pi \left[ -\frac{x^5}{5} - \frac{x^3}{3} + 3x^2 + 8x \right]_{-1}^{2}$$
9. **Hitung nilai batas atas ($x=2$):**
$$-\frac{2^5}{5} - \frac{2^3}{3} + 3(2^2) + 8(2) = -\frac{32}{5} - \frac{8}{3} + 12 + 16$$
10. **Hitung nilai batas bawah ($x=-1$):**
$$-\frac{(-1)^5}{5} - \frac{(-1)^3}{3} + 3(-1)^2 + 8(-1) = -\frac{-1}{5} - \frac{-1}{3} + 3 - 8 = \frac{1}{5} + \frac{1}{3} - 5$$
11. **Hitung selisih:**
$$\left(-\frac{32}{5} - \frac{8}{3} + 28\right) - \left(\frac{1}{5} + \frac{1}{3} - 5\right) = \left(-\frac{32}{5} - \frac{8}{3} + 28\right) - \left(\frac{1}{5} + \frac{1}{3} - 5\right)$$
12. **Gabungkan pecahan:**
$$-\frac{32}{5} - \frac{8}{3} = -\frac{96}{15} - \frac{40}{15} = -\frac{136}{15}$$
$$\frac{1}{5} + \frac{1}{3} = \frac{3}{15} + \frac{5}{15} = \frac{8}{15}$$
13. **Hitung total:**
$$\left(-\frac{136}{15} + 28\right) - \left(\frac{8}{15} - 5\right) = \left(-\frac{136}{15} + \frac{420}{15}\right) - \left(\frac{8}{15} - \frac{75}{15}\right) = \frac{284}{15} - \left(-\frac{67}{15}\right) = \frac{284}{15} + \frac{67}{15} = \frac{351}{15} = \frac{117}{5}$$
14. **Jadi volume benda putar adalah:**
$$V = \pi \times \frac{117}{5} = \frac{117}{5} \pi$$
**Jawaban akhir:** Volume benda putar adalah $$\boxed{\frac{117}{5} \pi}$$ satuan volume.
Volume Benda Putar A6C112
Step-by-step solutions with LaTeX - clean, fast, and student-friendly.