To find the number of integral coordinate points that lie in the interior of the region common to the circle \(x^2 + y^2 = 16\) and the parabola \(y^2 = 4x\), we will follow these steps:
### Step 1: Understand the equations
1. The equation of the circle is \(x^2 + y^2 = 16\), which has a center at \((0,0)\) and a radius of \(4\).
2. The equation of the parabola is \(y^2 = 4x\), which opens to the right.
### Step 2: Determine the boundaries
- For the circle, the interior condition is given by:
\[
x^2 + y^2 < 16
\]
- For the parabola, the interior condition is given by:
\[
y^2 < 4x
\]
### Step 3: Analyze the range of \(x\)
- The circle restricts \(x\) to the interval \([-4, 4]\).
- The parabola \(y^2 = 4x\) is only defined for \(x \geq 0\), so we further restrict \(x\) to the interval \([0, 4]\).
### Step 4: Check integer values of \(x\)
- The possible integer values for \(x\) in the interval \([0, 4]\) are \(0, 1, 2, 3, 4\).
### Step 5: Determine corresponding \(y\) values for each \(x\)
1. **For \(x = 0\)**:
- From the circle: \(0^2 + y^2 < 16 \Rightarrow y^2 < 16 \Rightarrow -4 < y < 4\) (possible \(y\) values: \(-3, -2, -1, 0, 1, 2, 3\) → 7 values)
- From the parabola: \(y^2 < 4 \cdot 0 \Rightarrow y^2 < 0\) (no valid \(y\) values)
- **Total points: 0**
2. **For \(x = 1\)**:
- From the circle: \(1^2 + y^2 < 16 \Rightarrow y^2 < 15 \Rightarrow -\sqrt{15} < y < \sqrt{15}\) (possible \(y\) values: \(-3, -2, -1, 0, 1, 2, 3\) → 7 values)
- From the parabola: \(y^2 < 4 \cdot 1 \Rightarrow y^2 < 4 \Rightarrow -2 < y < 2\) (possible \(y\) values: \(-1, 0, 1\) → 3 values)
- **Total points: 3**
3. **For \(x = 2\)**:
- From the circle: \(2^2 + y^2 < 16 \Rightarrow y^2 < 12 \Rightarrow -\sqrt{12} < y < \sqrt{12}\) (possible \(y\) values: \(-3, -2, -1, 0, 1, 2, 3\) → 7 values)
- From the parabola: \(y^2 < 4 \cdot 2 \Rightarrow y^2 < 8 \Rightarrow -\sqrt{8} < y < \sqrt{8}\) (possible \(y\) values: \(-2, -1, 0, 1, 2\) → 5 values)
- **Total points: 5**
4. **For \(x = 3\)**:
- From the circle: \(3^2 + y^2 < 16 \Rightarrow y^2 < 7 \Rightarrow -\sqrt{7} < y < \sqrt{7}\) (possible \(y\) values: \(-2, -1, 0, 1, 2\) → 5 values)
- From the parabola: \(y^2 < 4 \cdot 3 \Rightarrow y^2 < 12 \Rightarrow -\sqrt{12} < y < \sqrt{12}\) (possible \(y\) values: \(-3, -2, -1, 0, 1, 2, 3\) → 7 values)
- **Total points: 5**
5. **For \(x = 4\)**:
- From the circle: \(4^2 + y^2 < 16 \Rightarrow y^2 < 0\) (no valid \(y\) values)
- From the parabola: \(y^2 < 4 \cdot 4 \Rightarrow y^2 < 16 \Rightarrow -4 < y < 4\) (possible \(y\) values: \(-3, -2, -1, 0, 1, 2, 3\) → 7 values)
- **Total points: 0**
### Step 6: Sum the total points
- Total points from all valid \(x\) values:
- \(0 + 3 + 5 + 5 + 0 = 13\)
### Final Answer
The total number of points with integral coordinates that lie in the interior of the common region is **13**.
