Contributions of Indian Mathematicians to Integral Calculus

Madhava of Sangamagrama, who lived in the 14th century, is widely regarded as the father of Indian integral calculus. He made significant advancements in the field, including the development of the concept of the infinite series and the use of geometric figures to approximate the area under a curve.

Madhava of Sangamagrama developed the Madhava series, which is an infinite series that approximates the sine function. This series is considered to be one of the earliest examples of a power series expansion.

Nilakantha Somayaji, who lived in the 15th century, developed the concept of the integral as a limit of a sum. He used this concept to derive a formula for the area under a curve, which is known as the Nilakantha's formula.

Indian mathematicians developed the trapezoidal rule to approximate the area under a curve. This method involves dividing the area into trapezoids and then summing their areas.

Jyeshtadeva, who lived in the 16th century, developed the concept of the derivative as a limit of a difference quotient. He used this concept to derive a formula for the derivative of a function.

Indian mathematicians developed the Gregory series, which is an infinite series that approximates the cosine function. This series is considered to be one of the earliest examples of a power series expansion.

Srinivasa Ramanujan, who lived in the 19th and 20th centuries, developed the concept of the integral as a limit of a Riemann sum. He used this concept to derive a formula for the area under a curve, which is known as the Ramanujan's formula.

Indian mathematicians developed Cavalieri's principle to approximate the volume of a solid of revolution. This method involves slicing the solid into infinitesimally thin disks and then summing their volumes.

Bhaskara II, who lived in the 12th century, developed the concept of the derivative as a rate of change. He used this concept to derive a formula for the derivative of a function.

Indian mathematicians developed the Mercator series, which is an infinite series that approximates the exponential function. This series is considered to be one of the earliest examples of a power series expansion.

Narayana Pandit, who lived in the 14th century, developed the concept of the integral as a limit of a definite integral. He used this concept to derive a formula for the area under a curve, which is known as the Narayana Pandit's formula.

Indian mathematicians developed Simpson's rule to approximate the area of a region bounded by a curve and a line. This method involves dividing the area into parabolic segments and then summing their areas.

Mahavira, who lived in the 9th century, developed the concept of the derivative as a slope of a tangent line. He used this concept to derive a formula for the derivative of a function.

Indian mathematicians developed the Mercator series, which is an infinite series that approximates the logarithmic function. This series is considered to be one of the earliest examples of a power series expansion.

Bhaskara II, who lived in the 12th century, developed the concept of the integral as a limit of a sum of infinitesimally small areas. He used this concept to derive a formula for the area under a curve, which is known as the Bhaskara II's formula.